2000 character limit reached
Generalized Chevalley criteria in simplicial homotopy type theory
Published 13 Mar 2024 in math.CT, cs.LO, math.AT, and math.LO | (2403.08190v1)
Abstract: We provide a generalized treatment of (co)cartesian arrows, fibrations, and functors. Compared to the classical conditions, the endpoint inclusions get replaced by arbitrary shape inclusions. Our framework is Riehl--Shulman's simplicial homotopy type theory which supports the development of synthetic internal $(\infty,1)$-category theory.
- Steve Awodey and Michael A. Warren “Homotopy theoretic models of identity types” In Mathematical Proceedings of the Cambridge Philosophical Society 146.1 Cambridge University Press, 2009, pp. 45–55 DOI: 10.1017/S0305004108001783
- “Fibrations of ∞\infty∞-categories” In Higher Structures 4.1, 2020 URL: http://journals.mq.edu.au/index.php/higher_structures/article/view/29
- César Bardomiano Martínez “Limits and exponentiable functors in simplicial homotopy type theory”, 2022 URL: https://arxiv.org/abs/2202.12386
- “Fibrations in ∞\infty∞-category theory” In 2016 MATRIX annals Cham: Springer, 2018, pp. 17–42 DOI: 10.1007/978-3-319-72299-3_2
- “Synthetic fibered (∞,1)1(\infty,1)( ∞ , 1 )-category theory” In Higher Structures 7, 2023, pp. 74–165 DOI: 10.21136/HS.2023.04
- “Cubical Type Theory: a constructive interpretation of the univalence axiom” In 21st International Conference on Types for Proofs and Programs (TYPES 2015), LIPIcs. Leibniz Int. Proc. Inform. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2018 DOI: 10.4230/LIPIcs.TYPES.2015.5
- “Formal Category Theory: A Course Held at Masaryk University”, 2018
- John W. Gray “Fibred and Cofibred Categories” In Proceedings of the Conference on Categorical Algebra Berlin, Heidelberg: Springer Berlin Heidelberg, 1966, pp. 21–83 DOI: https://doi.org/10.1007/978-3-642-99902-4_2
- “An (∞,2)2(\infty,2)( ∞ , 2 )-categorical pasting theorem”, 2021 arXiv:2106.03660
- “A 2222-categorical proof of Frobenius for fibrations defined from a generic point”, 2022 URL: https://arxiv.org/abs/2210.00078
- André Joyal “Quasi-categories and Kan complexes” Special volume celebrating the 70th birthday of Professor Max Kelly In J. Pure Appl. Algebra 175.1-3, 2002, pp. 207–222 DOI: 10.1016/S0022-4049(02)00135-4
- Nikolai Kudasov “Rzk” An experimental proof assistant based on a type theory for synthetic ∞\infty∞-categories URL: https://github.com/rzk-lang/rzk
- Nikolai Kudasov, Emily Riehl and Jonathan Weinberger “Formalizing the ∞\infty∞-Categorical Yoneda Lemma” In Proceedings of the 13th ACM SIGPLAN International Conference on Certified Programs and Proofs, 2024, pp. 274–290 DOI: 10.1145/3636501.3636945
- Fosco Loregian “(Co) end calculus” Cambridge University Press, 2021
- “Categorical notions of fibration” In Expo. Math. 38.4, 2020, pp. 496–514 DOI: 10.1016/j.exmath.2019.02.004
- Jacob Lurie “Higher Topos Theory”, Annals of Mathematics Studies 170 Princeton University Press, 2009 arXiv:math/0608040
- Louis Martini “Cocartesian fibrations and straightening internal to an ∞\infty∞-topos” arXiv, 2022 URL: https://arxiv.org/abs/2204.00295
- “Internal higher topos theory”, 2023 arXiv: https://arxiv.org/abs/2303.06437
- Ian Orton and Andrew M. Pitts “Axioms for modelling cubical type theory in a topos” Id/No 24 In 25th EACSL annual conference and 30th workshop on computer science logic, CSL’16, Marseille, France, August 29 – September 1, 2016. Proceedings Wadern: Schloss Dagstuhl – Leibniz Zentrum für Informatik, 2016, pp. 19 DOI: 10.4230/LIPIcs.CSL.2016.24
- Nima Rasekh “Cartesian Fibrations of Complete Segal Spaces” In Higher Structures 7, 2023, pp. 40–73 DOI: https://articles.math.cas.cz/10.21136/HS.2023.03
- Emily Riehl “Could ∞\infty∞-Category Theory Be Taught to Undergraduates?” In Notices of the American Mathematical Society 70.5, 2023 DOI: 10.1090/noti2692
- Emily Riehl “On the ∞\infty∞-topos semantics of homotopy type theory” In Bulletin of the London Mathematical Society 56.2, 2024, pp. 461–517 DOI: 10.1112/blms.12997
- “A type theory for synthetic ∞\infty∞-categories” In Higher Structures 1.1, 2017, pp. 147–224 URL: https://higher-structures.math.cas.cz/api/files/issues/Vol1Iss1/RiehlShulman
- “Elements of ∞\infty∞-Category Theory”, Cambridge Studies in Advanced Mathematics Cambridge University Press, 2022 URL: https://emilyriehl.github.io/files/elements.pdf
- Egbert Rijke “Introduction to Homotopy Type Theory” To appear at Cambridge University Press, 2022 URL: https://arxiv.org/abs/2212.11082
- The sHoTT Community “sHoTT Library in Rzk”, 2024 URL: https://rzk-lang.github.io/sHoTT/
- Michael Shulman “All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes”, 2019 arXiv: https://arxiv.org/abs/1904.07004
- Ross Street “Fibrations and Yoneda’s lemma in a 2222-category” In Category Seminar (Proc. Sem., Sydney, 1972/1973), 1974, pp. 104–133. Lecture Notes in Math.\bibrangessepVol. 420 DOI: 10.1007/BFb0063102
- The Univalent Foundations Program “Homotopy Type Theory: Univalent Foundations of Mathematics” Institute for Advanced Study: https://homotopytypetheory.org/book, 2013
- Friedrich Ulmer “Properties of dense and relative adjoint functors” In J. Algebra 8, 1968, pp. 77–95 DOI: 10.1016/0021-8693(68)90036-7
- Matthew Z. Weaver and Daniel R. Licata “A Constructive Model of Directed Univalence in Bicubical Sets” In Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS ’20 Saarbrücken, Germany: Association for Computing Machinery, 2020, pp. 915–928 DOI: 10.1145/3373718.3394794
- Jonathan Weinberger “A Synthetic Perspective on (∞,1)1(\infty,1)( ∞ , 1 )-Category Theory: Fibrational and Semantic Aspects”, 2022, pp. xxi+177 DOI: https://doi.org/10.26083/tuprints-00020716
- Jonathan Weinberger “Internal sums for synthetic fibered (∞,1)1(\infty,1)( ∞ , 1 )-categories” To appear in Journal of Pure and Applied Algebra, 2022 URL: https://arxiv.org/pdf/2205.00386.pdf
- Jonathan Weinberger “Strict stability of extension types” arXiv, 2022 DOI: 10.48550/ARXIV.2203.07194
- Jonathan Weinberger “Two-sided cartesian fibrations of synthetic (∞,1)1(\infty,1)( ∞ , 1 )-categories”, 2022 arXiv: https://arxiv.org/abs/2204.00938
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.