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Computational Synthetic Cohomology Theory in Homotopy Type Theory (2401.16336v2)

Published 29 Jan 2024 in math.AT and cs.LO

Abstract: This paper discusses the development of synthetic cohomology in Homotopy Type Theory (HoTT), as well as its computer formalisation. The objectives of this paper are (1) to generalise previous work on integral cohomology in HoTT by the current authors and Brunerie (2022) to cohomology with arbitrary coefficients and (2) to provide the mathematical details of, as well as extend, results underpinning the computer formalisation of cohomology rings by the current authors and Lamiaux (2023). With respect to objective (1), we provide new direct definitions of the cohomology group operations and of the cup product, which, just as in (Brunerie et al., 2022), enable significant simplifications of many earlier proofs in synthetic cohomology theory. In particular, the new definition of the cup product allows us to give the first complete formalisation of the axioms needed to turn the cohomology groups into a graded commutative ring. We also establish that this cohomology theory satisfies the HoTT formulation of the Eilenberg-Steenrod axioms for cohomology and study the classical Mayer-Vietoris and Gysin sequences. With respect to objective (2), we characterise the cohomology groups and rings of various spaces, including the spheres, torus, Klein bottle, real/complex projective planes, and infinite real projective space. All results have been formalised in Cubical Agda and we obtain multiple new numbers, similar to the famous `Brunerie number', which can be used as benchmarks for computational implementations of HoTT. Some of these numbers are infeasible to compute in Cubical Agda and hence provide new computational challenges and open problems which are much easier to define than the original Brunerie number.

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References (40)
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URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. 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In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Ulrik Buchholtz, Floris van Doorn, and Egbert Rijke. Higher Groups in Homotopy Type Theory. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’18, pages 205–214, New York, NY, USA, 2018. Association for Computing Machinery. ISBN 9781450355834. doi: 10.1145/3209108.3209150. URL https://doi.org/10.1145/3209108.3209150. Buchholtz et al. [2023] Ulrik Buchholtz, J. Daniel Christensen, Jarl G. Taxerås Flaten, and Egbert Rijke. Central H-spaces and banded types, 2023. Cano et al. [2016] Guillaume Cano, Cyril Cohen, Maxime Dénès, Anders Mörtberg, and Vincent Siles. Formalized Linear Algebra over Elementary Divisor Rings in Coq. Logical Methods in Computer Science, 12(2), 2016. doi: 10.2168/LMCS-12(2:7)2016. URL http://dx.doi.org/10.2168/LMCS-12(2:7)2016. Cavallo [2015] Evan Cavallo. Synthetic Cohomology in Homotopy Type Theory. Master’s thesis, Carnegie Mellon University, 2015. Cavallo [2022] Evan Cavallo. Formalization of the Evan’s Trick, 2022. URL https://github.com/agda/cubical/blob/f8d6fcf3bc24ce3915b3960ad3b716fd082e8b9d/Cubical/Foundations/Pointed/Homogeneous.agda#L42. Christensen and Flaten [2023] J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Ulrik Buchholtz, J. Daniel Christensen, Jarl G. Taxerås Flaten, and Egbert Rijke. Central H-spaces and banded types, 2023. Cano et al. [2016] Guillaume Cano, Cyril Cohen, Maxime Dénès, Anders Mörtberg, and Vincent Siles. Formalized Linear Algebra over Elementary Divisor Rings in Coq. Logical Methods in Computer Science, 12(2), 2016. doi: 10.2168/LMCS-12(2:7)2016. URL http://dx.doi.org/10.2168/LMCS-12(2:7)2016. Cavallo [2015] Evan Cavallo. Synthetic Cohomology in Homotopy Type Theory. Master’s thesis, Carnegie Mellon University, 2015. Cavallo [2022] Evan Cavallo. Formalization of the Evan’s Trick, 2022. URL https://github.com/agda/cubical/blob/f8d6fcf3bc24ce3915b3960ad3b716fd082e8b9d/Cubical/Foundations/Pointed/Homogeneous.agda#L42. Christensen and Flaten [2023] J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Guillaume Cano, Cyril Cohen, Maxime Dénès, Anders Mörtberg, and Vincent Siles. Formalized Linear Algebra over Elementary Divisor Rings in Coq. Logical Methods in Computer Science, 12(2), 2016. doi: 10.2168/LMCS-12(2:7)2016. URL http://dx.doi.org/10.2168/LMCS-12(2:7)2016. Cavallo [2015] Evan Cavallo. Synthetic Cohomology in Homotopy Type Theory. Master’s thesis, Carnegie Mellon University, 2015. Cavallo [2022] Evan Cavallo. Formalization of the Evan’s Trick, 2022. URL https://github.com/agda/cubical/blob/f8d6fcf3bc24ce3915b3960ad3b716fd082e8b9d/Cubical/Foundations/Pointed/Homogeneous.agda#L42. Christensen and Flaten [2023] J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Evan Cavallo. Synthetic Cohomology in Homotopy Type Theory. Master’s thesis, Carnegie Mellon University, 2015. Cavallo [2022] Evan Cavallo. Formalization of the Evan’s Trick, 2022. URL https://github.com/agda/cubical/blob/f8d6fcf3bc24ce3915b3960ad3b716fd082e8b9d/Cubical/Foundations/Pointed/Homogeneous.agda#L42. Christensen and Flaten [2023] J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Evan Cavallo. Formalization of the Evan’s Trick, 2022. URL https://github.com/agda/cubical/blob/f8d6fcf3bc24ce3915b3960ad3b716fd082e8b9d/Cubical/Foundations/Pointed/Homogeneous.agda#L42. Christensen and Flaten [2023] J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. 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Synthetic Cohomology in Homotopy Type Theory. Master’s thesis, Carnegie Mellon University, 2015. Cavallo [2022] Evan Cavallo. Formalization of the Evan’s Trick, 2022. URL https://github.com/agda/cubical/blob/f8d6fcf3bc24ce3915b3960ad3b716fd082e8b9d/Cubical/Foundations/Pointed/Homogeneous.agda#L42. Christensen and Flaten [2023] J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Guillaume Cano, Cyril Cohen, Maxime Dénès, Anders Mörtberg, and Vincent Siles. Formalized Linear Algebra over Elementary Divisor Rings in Coq. Logical Methods in Computer Science, 12(2), 2016. doi: 10.2168/LMCS-12(2:7)2016. URL http://dx.doi.org/10.2168/LMCS-12(2:7)2016. Cavallo [2015] Evan Cavallo. Synthetic Cohomology in Homotopy Type Theory. Master’s thesis, Carnegie Mellon University, 2015. Cavallo [2022] Evan Cavallo. Formalization of the Evan’s Trick, 2022. URL https://github.com/agda/cubical/blob/f8d6fcf3bc24ce3915b3960ad3b716fd082e8b9d/Cubical/Foundations/Pointed/Homogeneous.agda#L42. Christensen and Flaten [2023] J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Evan Cavallo. Synthetic Cohomology in Homotopy Type Theory. Master’s thesis, Carnegie Mellon University, 2015. Cavallo [2022] Evan Cavallo. Formalization of the Evan’s Trick, 2022. URL https://github.com/agda/cubical/blob/f8d6fcf3bc24ce3915b3960ad3b716fd082e8b9d/Cubical/Foundations/Pointed/Homogeneous.agda#L42. Christensen and Flaten [2023] J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Evan Cavallo. Formalization of the Evan’s Trick, 2022. URL https://github.com/agda/cubical/blob/f8d6fcf3bc24ce3915b3960ad3b716fd082e8b9d/Cubical/Foundations/Pointed/Homogeneous.agda#L42. Christensen and Flaten [2023] J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. 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Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Guillaume Cano, Cyril Cohen, Maxime Dénès, Anders Mörtberg, and Vincent Siles. Formalized Linear Algebra over Elementary Divisor Rings in Coq. Logical Methods in Computer Science, 12(2), 2016. doi: 10.2168/LMCS-12(2:7)2016. URL http://dx.doi.org/10.2168/LMCS-12(2:7)2016. Cavallo [2015] Evan Cavallo. Synthetic Cohomology in Homotopy Type Theory. Master’s thesis, Carnegie Mellon University, 2015. Cavallo [2022] Evan Cavallo. Formalization of the Evan’s Trick, 2022. URL https://github.com/agda/cubical/blob/f8d6fcf3bc24ce3915b3960ad3b716fd082e8b9d/Cubical/Foundations/Pointed/Homogeneous.agda#L42. Christensen and Flaten [2023] J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Evan Cavallo. Synthetic Cohomology in Homotopy Type Theory. Master’s thesis, Carnegie Mellon University, 2015. Cavallo [2022] Evan Cavallo. Formalization of the Evan’s Trick, 2022. URL https://github.com/agda/cubical/blob/f8d6fcf3bc24ce3915b3960ad3b716fd082e8b9d/Cubical/Foundations/Pointed/Homogeneous.agda#L42. Christensen and Flaten [2023] J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Evan Cavallo. Formalization of the Evan’s Trick, 2022. URL https://github.com/agda/cubical/blob/f8d6fcf3bc24ce3915b3960ad3b716fd082e8b9d/Cubical/Foundations/Pointed/Homogeneous.agda#L42. Christensen and Flaten [2023] J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
  11. Formalized Linear Algebra over Elementary Divisor Rings in Coq. Logical Methods in Computer Science, 12(2), 2016. doi: 10.2168/LMCS-12(2:7)2016. URL http://dx.doi.org/10.2168/LMCS-12(2:7)2016. Cavallo [2015] Evan Cavallo. Synthetic Cohomology in Homotopy Type Theory. Master’s thesis, Carnegie Mellon University, 2015. Cavallo [2022] Evan Cavallo. Formalization of the Evan’s Trick, 2022. URL https://github.com/agda/cubical/blob/f8d6fcf3bc24ce3915b3960ad3b716fd082e8b9d/Cubical/Foundations/Pointed/Homogeneous.agda#L42. Christensen and Flaten [2023] J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Evan Cavallo. Synthetic Cohomology in Homotopy Type Theory. Master’s thesis, Carnegie Mellon University, 2015. Cavallo [2022] Evan Cavallo. Formalization of the Evan’s Trick, 2022. URL https://github.com/agda/cubical/blob/f8d6fcf3bc24ce3915b3960ad3b716fd082e8b9d/Cubical/Foundations/Pointed/Homogeneous.agda#L42. Christensen and Flaten [2023] J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Evan Cavallo. Formalization of the Evan’s Trick, 2022. URL https://github.com/agda/cubical/blob/f8d6fcf3bc24ce3915b3960ad3b716fd082e8b9d/Cubical/Foundations/Pointed/Homogeneous.agda#L42. Christensen and Flaten [2023] J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. 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Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. 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Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. 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Christensen and Flaten [2023] J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. 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Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. 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Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
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Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
  14. J. Daniel Christensen and Jarl G. Taxerås Flaten. Ext groups in homotopy type theory, 2023. Christensen and Scoccola [2023] J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
  15. The Hurewicz theorem in Homotopy Type Theory. Algebraic & Geometric Topology, 23:2107–2140, 2023. Eilenberg and Steenrod [1952] Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Samuel Eilenberg and Norman Steenrod. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
  16. Foundations of Algebraic Topology. Foundations of Algebraic Topology. Princeton University Press, 1952. ISBN 9780608102351. Graham [2018] Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
  17. Robert Graham. Synthetic Homology in Homotopy Type Theory, 2018. URL https://arxiv.org/abs/1706.01540. Preprint. Harington [2020] Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Elies Harington. Groupes de cohomologie en théorie des types univalente, 2020. Internship report, supervised by Thierry Coquand. Hatcher [2002] Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
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URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. 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In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
  19. Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 9780521795401. URL https://pi.math.cornell.edu/~hatcher/AT/AT.pdf. Heras et al. [2012] Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Maxime Dénès, Gadea Mata, Anders Mörtberg, María Poza, and Vincent Siles. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
  20. Towards a Certified Computation of Homology Groups for Digital Images. In Proceedings of the 4th International Conference on Computational Topology in Image Context, CTIC’12, pages 49–57, Berlin, Heidelberg, 2012. Springer-Verlag. ISBN 9783642302374. doi: 10.1007/978-3-642-30238-1˙6. URL https://doi.org/10.1007/978-3-642-30238-1_6. Heras et al. [2013] Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Jónathan Heras, Thierry Coquand, Anders Mörtberg, and Vincent Siles. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
  21. Computing Persistent Homology Within Coq/SSReflect. ACM Transactions on Computational Logic, 14(4):1–26, 2013. ISSN 1529-3785. doi: 10.1145/2528929. URL http://doi.acm.org/10.1145/2528929. Hou et al.(2016)Hou (Favonia), Finster, Licata, and Lumsdaine [Favonia]Kuen-Bang Hou (Favonia), Eric Finster, Daniel R. Licata, and Peter LeFanu Lumsdaine. A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’16, pages 565–574, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4391-6. doi: 10.1145/2933575.2934545. Jack [2023] Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. 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Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. 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The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Tom Jack. π4⁢(𝕊3)≇1subscript𝜋4superscript𝕊31\pi_{4}(\mathbb{S}^{3})\not\cong 1italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( blackboard_S start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≇ 1 1 and another brunerie number in cchm. The Second International Conference on Homotopy Type Theory (HoTT 2023), 2023. URL https://hott.github.io/HoTT-2023/abstracts/HoTT-2023_abstract_21.pdf. Lamiaux et al. [2023] Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
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Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Thomas Lamiaux, Axel Ljungström, and Anders Mörtberg. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
  23. Computing cohomology rings in cubical agda. In Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2023, pages 239–252, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400700262. doi: 10.1145/3573105.3575677. URL https://doi.org/10.1145/3573105.3575677. Licata and Brunerie [2014] Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. 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In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
  24. A cubical type theory, November 2014. URL http://dlicata.web.wesleyan.edu/pubs/lb14cubical/lb14cubes-oxford.pdf. Talk at Oxford Homotopy Type Theory Workshop. Licata and Brunerie [2015] Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Guillaume Brunerie. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
  25. A Cubical Approach to Synthetic Homotopy Theory. In Proceedings of the 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’15, pages 92–103, Washington, DC, USA, 2015. IEEE Computer Society. ISBN 978-1-4799-8875-4. doi: 10.1109/LICS.2015.19. Licata and Finster [2014] Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Daniel R. Licata and Eric Finster. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
  26. Eilenberg-MacLane Spaces in Homotopy Type Theory. In Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, New York, NY, USA, 2014. Association for Computing Machinery. ISBN 9781450328869. doi: 10.1145/2603088.2603153. URL https://doi.org/10.1145/2603088.2603153. Ljungström [2023] Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
  27. Axel Ljungström. Symmetric monoidal smash products in hott, 2023. URL https://aljungstrom.github.io/files/smash.pdf. Unpublished note accompanying talk at Homotopy Type Theory 2023, Carnegie Mellon University, Pittsburgh, USA. Ljungström and Mörtberg [2023] Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Axel Ljungström and Anders Mörtberg. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
  28. Formalizing π4⁢(s3)≅ℤ/2⁢ℤsubscript𝜋4superscript𝑠3ℤ2ℤ\pi_{4}(s^{3})\cong\mathbb{Z}/2\mathbb{Z}italic_π start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_s start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≅ blackboard_Z / 2 blackboard_Z and computing a brunerie number in cubical agda. In 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). IEEE, June 2023. doi: 10.1109/lics56636.2023.10175833. URL http://dx.doi.org/10.1109/LICS56636.2023.10175833. mathlib Community [2020] The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
  29. The mathlib Community. The lean mathematical library. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 367–381, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373824. URL https://doi.org/10.1145/3372885.3373824. Mörtberg and Pujet [2020] Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Anders Mörtberg and Loïc Pujet. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
  30. Cubical Synthetic Homotopy Theory. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, CPP 2020, pages 158–171, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450370974. doi: 10.1145/3372885.3373825. URL https://doi.org/10.1145/3372885.3373825. Qian [2019] Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Zesen Qian. Towards Eilenberg-MacLane Spaces in Cubical Type Theory. Master’s thesis, Carnegie Mellon University, 2019. Shulman [2013] Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
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Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
  32. Michael Shulman. Cohomology, 2013. post on the Homotopy Type Theory blog: http://homotopytypetheory.org/2013/07/24/. Shulman [2019] Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Michael Shulman. All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes, April 2019. URL https://arxiv.org/abs/1904.07004. Preprint. Sojakova [2016] Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.
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  34. Kristina Sojakova. The Equivalence of the Torus and the Product of Two Circles in Homotopy Type Theory. ACM Transactions on Computational Logic, 17(4):29:1–29:19, November 2016. ISSN 1529-3785. doi: 10.1145/2992783. The Coq Development Team [2021] The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Coq Development Team. The Coq Proof Assistant, 2021. URL https://www.coq.inria.fr. The Spectral Sequence Project [2018] The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Spectral Sequence Project, 2018. https://github.com/cmu-phil/Spectral. The Univalent Foundations Program [2013] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Self-published, Institute for Advanced Study, 2013. URL https://homotopytypetheory.org/book/. van Doorn [2018] Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Floris van Doorn. On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory. PhD thesis, Carnegie Mellon University, May 2018. URL https://arxiv.org/abs/1808.10690. Vezzosi et al. [2021] Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. Andrea Vezzosi, Anders Mörtberg, and Andreas Abel. Cubical agda: A dependently typed programming language with univalence and higher inductive types. Journal of Functional Programming, 31:e8, 2021. doi: 10.1017/S0956796821000034. Wärn [2023] David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. 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  40. David Wärn. Eilenberg–Maclane spaces and stabilisation in homotopy type theory. Journal of Homotopy and Related Structures, 18(2):357–368, Sep 2023. ISSN 1512-2891. doi: 10.1007/s40062-023-00330-5. URL https://doi.org/10.1007/s40062-023-00330-5.

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