Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
156 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Two-sided cartesian fibrations of synthetic $(\infty,1)$-categories (2204.00938v3)

Published 2 Apr 2022 in math.CT, cs.LO, math.AT, and math.LO

Abstract: Within the framework of Riehl-Shulman's synthetic $(\infty,1)$-category theory, we present a theory of two-sided cartesian fibrations. Central results are several characterizations of the two-sidedness condition `{a} la Chevalley, Gray, Street, and Riehl-Verity, a two-sided Yoneda Lemma, as well as the proof of several closure properties. Along the way, we also define and investigate a notion of fibered or sliced fibration which is used later to develop the two-sided case in a modular fashion. We also briefly discuss discrete two-sided cartesian fibrations in this setting, corresponding to $(\infty,1)$-distributors. The systematics of our definitions and results closely follows Riehl-Verity's $\infty$-cosmos theory, but formulated internally to Riehl-Shulman's simplicial extension of homotopy type theory. All the constructions and proofs in this framework are by design invariant under homotopy equivalence. Semantically, the synthetic $(\infty,1)$-categories correspond to internal $(\infty,1)$-categories implemented as Rezk objects in an arbitrary given $(\infty,1)$-topos.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (100)
  1. Benedikt Ahrens, Krzysztof Kapulkin and Michael Shulman “Univalent categories and the Rezk completion” In Math. Struct. Comput. Sci. 25.5, 2015, pp. 1010–1039 DOI: 10.1017/S0960129514000486
  2. “The Univalence Principle” arXiv, 2021 DOI: 10.48550/ARXIV.2102.06275
  3. Benedikt Ahrens, Paige Randall North and Niels Weide “Bicategorical type theory: semantics and syntax” In Mathematical Structures in Computer Science 33.10, 2023, pp. 868–912 DOI: 10.1017/S0960129523000312
  4. “Two-Level Type Theory and Applications”, 2019 arXiv: https://arxiv.org/abs/1705.03307
  5. Steve Awodey “Type theory and homotopy” arXiv, 2010 DOI: 10.48550/ARXIV.1010.1810
  6. 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
  7. “Fibrations of ∞\infty∞-categories” In Higher Structures 4.1, 2020 URL: http://journals.mq.edu.au/index.php/higher_structures/article/view/29
  8. Igor Bakovic “Fibrations of bicategories” In Preprint available at http://www. irb. hr/korisnici/ibakovic/groth2fib. pdf, 2011
  9. César Bardomiano Martínez “Limits and exponentiable functors in simplicial homotopy type theory”, 2022 arXiv:2202.12386 [math.CT]
  10. “Parametrized higher category theory and higher algebra: Exposé I – Elements of parametrized higher category theory”, 2016 arXiv: https://arxiv.org/abs/1608.03657
  11. “Fibrations in ∞\infty∞-category theory” In 2016 MATRIX annals Cham: Springer, 2018, pp. 17–42 DOI: 10.1007/978-3-319-72299-3˙2
  12. Jean Bénabou “Distributors at Work” Notes from lectures at TU Darmstadt taken by Thomas Streicher, 2000 URL: https://www2.mathematik.tu-darmstadt.de/~streicher/FIBR/DiWo.pdf
  13. Jean Bénabou “Les distributeurs, Rapport 33, 1973, Inst. de Math” In Pure et Appl. Univ. Cath. Louvain la Neuve
  14. David Li-Bland “The stack of higher internal categories and stacks of iterated spans”, 2015 arXiv: https://arxiv.org/abs/1506.08870
  15. Pedro Boavida de Brito “Segal objects and the Grothendieck construction” In An alpine bouquet of algebraic topology 708, Contemp. Math. Amer. Math. Soc., [Providence], RI, 2018, pp. 19–44 DOI: 10.1090/conm/708/14271
  16. Francis Borceux “Handbook of Categorical Algebra: Volume 2, Categories and Structures” Cambridge University Press, 1994
  17. Ulrik Buchholtz “Higher Structures in Homotopy Type Theory” In Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts Cham: Springer International Publishing, 2019, pp. 151–172 DOI: 10.1007/978-3-030-15655-8˙7
  18. “Synthetic fibered (∞,1)1(\infty,1)( ∞ , 1 )-category theory” In Higher Structures 7, 2023, pp. 74–165 DOI: 10.21136/HS.2023.04
  19. Paolo Capriotti “Models of Type Theory with Strict Equality”, 2016 URL: http://arxiv.org/abs/1702.04912
  20. “Univalent Higher Categories via Complete Semi-Segal Types” In Proc. ACM Program. Lang. 2.POPL New York, NY, USA: Association for Computing Machinery, 2017 DOI: 10.1145/3158132
  21. Evan Cavallo, Emily Riehl and Christian Sattler “On the directed univalence axiom” Talk at AMS Special Session on Homotopy Type Theory, Joint Mathematics Meething, San Diego, 2018 URL: http://www.math.jhu.edu/~eriehl/JMM2018-directed-univalence.pdf
  22. “Fibered aspects of Yoneda’s regular span” In Advances in Mathematics 360, 2020, pp. 106899 DOI: https://doi.org/10.1016/j.aim.2019.106899
  23. Denis-Charles Cisinski “Higher Categories and Homotopical Algebra”, Cambridge Studies in Advanced Mathematics Cambridge University Press, 2019 DOI: 10.1017/9781108588737
  24. Denis-Charles Cisinski “Univalent universes for elegant models of homotopy types”, 2014 arXiv: https://arxiv.org/pdf/1406.0058.pdf
  25. Denis-Charles Cisinski and Hoang Kim Nguyen “The universal coCartesian fibration”, 2022 arXiv:2210.08945 [math.CT]
  26. Maria Manuel Clementino and Fernando Lucatelli Nunes “Lax comma 2222-categories and admissible 2222-functors”, 2020 arXiv: https://arxiv.org/abs/2002.03132
  27. “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
  28. G.S.H. Cruttwell and Michael A. Shulman “A unified framework for generalized multicategories” In Theory Appl. Categ. 24, 2010, pp. No. 21\bibrangessep580–655
  29. Ivan Di Liberti and Fosco Loregian “On the unicity of formal category theories” In arXiv preprint arXiv:1901.01594, 2019
  30. Imma Gálvez-Carrillo, Joachim Kock and Andrew Tonks “Decomposition spaces and restriction species” In International Mathematics Research Notices 2020.21 Oxford University Press, 2020, pp. 7558–7616
  31. David Gepner, Rune Haugseng and Thomas Nikolaus “Lax Colimits and Free Fibrations in ∞\infty∞-Categories” In Doc. Math. 22, 2017, pp. 1225–1266 DOI: 10.25537/dm.2017v22.1225-1266
  32. 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
  33. Daniel Grayson “An introduction to univalent foundations for mathematicians” In Bulletin of the American Mathematical Society 55.4 American Mathematical Society (AMS), 2018, pp. 427–450 DOI: 10.1090/bull/1616
  34. Rune Haugseng “The higher Morita category of 𝔼nsubscript𝔼𝑛\mathbb{E}_{n}blackboard_E start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT-algebras” In Geometry & Topology 21.3 Mathematical Sciences Publishers, 2017, pp. 1631–1730
  35. “Two-variable fibrations, factorisation systems and-categories of spans” In Forum of Mathematics, Sigma 11, 2023, pp. e111 Cambridge University Press
  36. Claudio Hermida “On fibred adjunctions and completeness for fibred categories” In Recent Trends in Data Type Specification Springer, 1992, pp. 235–251 DOI: 10.1007/3-540-57867-6˙14
  37. “The groupoid model refutes uniqueness of identity proofs” In Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science, 1994, pp. 208–212 DOI: 10.1109/LICS.1994.316071
  38. “Lifting Grothendieck universes” In Unpublished note, 199? URL: https://www2.mathematik.tu-darmstadt.de/~streicher/NOTES/lift.pdf
  39. André Joyal “Notes on quasi-categories”, 2008 URL: http://www.math.uchicago.edu/~may/IMA/Joyal.pdf
  40. 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
  41. Krzysztof Kapulkin and Peter LeFanu Lumsdaine “The simplicial model of Univalent Foundations (after Voevodsky)” In Journal of the European Mathematical Society 23.6, 2021, pp. 2071–2126
  42. Alex Kavvos “A Quantum of Direction” preprint, 2019 URL: https://www.lambdabetaeta.eu/papers/meio.pdf
  43. “Local fibred right adjoints are polynomial” In Math. Struct. Comput. Sci. 23.1 Cambridge University Press, 2013, pp. 131–141 DOI: 10.1017/S0960129512000217
  44. Nikolai Kudasov “Rzk” An experimental proof assistant based on a type theory for synthetic ∞\infty∞-categories URL: https://github.com/rzk-lang/rzk
  45. 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
  46. Daniel R. Licata and Robert Harper “2-dimensional directed type theory” In Twenty-Seventh Conference on the Mathematical Foundations of Programming Semantics (MFPS XXVII) 276, Electron. Notes Theor. Comput. Sci. Elsevier Sci. B. V., Amsterdam, 2011, pp. 263–289 DOI: 10.1016/j.entcs.2011.09.026
  47. “Categorical notions of fibration” In Expo. Math. 38.4, 2020, pp. 496–514 DOI: 10.1016/j.exmath.2019.02.004
  48. Jacob Lurie “Higher Algebra” https://www.math.ias.edu/~lurie/papers/HA.pdf, 2017
  49. Jacob Lurie “Higher Topos Theory”, Annals of Mathematics Studies 170 Princeton University Press, 2009 arXiv:math/0608040
  50. Louis Martini “Cocartesian fibrations and straightening internal to an ∞\infty∞-topos” In arXiv preprint arXiv:2204.00295, 2022
  51. Louis Martini “Yoneda’s lemma for internal higher categoriess”, 2021 arXiv: https://arxiv.org/abs/2103.17141
  52. “Internal higher topos theory” In arXiv preprint arXiv:2303.06437, 2023
  53. “Limits and colimits in internal higher category theory”, 2022 arXiv: https://arxiv.org/abs/2111.14495
  54. Erin McCloskey “Relative Weak Factorization Systems”, 2022
  55. Giuseppe Metere “Distributors and the comprehensive factorization system for internal groupoids” In arXiv preprint arXiv:1701.05139, 2017
  56. Hoang Kim Nguyen “Theorems in Higher Category Theory and Applications”, 2019 URL: https://epub.uni-regensburg.de/38448/
  57. Paige Randall North “Towards a Directed Homotopy Type Theory” Proceedings of the Thirty-Fifth Conference on the Mathematical Foundations of Programming Semantics In Electronic Notes in Theoretical Computer Science 347, 2019, pp. 223–239 DOI: https://doi.org/10.1016/j.entcs.2019.09.012
  58. Andreas Nuyts “Towards a Directed Homotopy Type Theory based on 4 Kinds of Variance”, 2015 URL: https://anuyts.github.io/files/mathesis.pdf
  59. 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
  60. “2-catégories réductibles” In Theory and Applications of Categories Reprints (no. 19, 2010), 1978
  61. Nima Rasekh “Cartesian fibrations and representability” In Homology, Homotopy and Applications 24.2 International Press of Boston, 2022, pp. 135–161
  62. 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
  63. Charles Rezk “Stuff about quasicategories”, 2017 URL: https://faculty.math.illinois.edu/~rezk/quasicats.pdf
  64. Charles Rezk “Toposes and homotopy toposes” Unpublished note. http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf, 2010
  65. Emily Riehl “Math 721: Homotopy type theory” Course notes, 2021 URL: https://github.com/emilyriehl/721/blob/master/721lectures.pdf
  66. “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
  67. “Cartesian exponentiation and monadicity”, 2021 arXiv: https://arxiv.org/abs/2101.09853
  68. “Elements of ∞\infty∞-Category Theory”, Cambridge Studies in Advanced Mathematics Cambridge University Press, 2022
  69. “Fibrations and Yoneda’s lemma in an ∞\infty∞-cosmos” In J. Pure Appl. Algebra 221.3, 2017, pp. 499–564 DOI: 10.1016/j.jpaa.2016.07.003
  70. “Homotopy coherent adjunctions and the formal theory of monads” In Advances in Mathematics 286, 2016, pp. 802–888 DOI: https://doi.org/10.1016/j.aim.2015.09.011
  71. “Infinity category theory from scratch” In Higher Structures 4.1, 2020 URL: https://higher-structures.math.cas.cz/api/files/issues/Vol4Iss1/RiehlVerity
  72. “Kan extensions and the calculus of modules for ∞\infty∞-categories” In Algebr. Geom. Topol. 17.1, 2017, pp. 189–271 DOI: 10.2140/agt.2017.17.189
  73. “The 2222-category theory of quasi-categories” In Advances in Mathematics 280, 2015, pp. 549–642 DOI: 10.1016/j.aim.2015.04.021
  74. Egbert Rijke “Introduction to Homotopy Type Theory”, 2020 URL: https://github.com/EgbertRijke/HoTT-Intro
  75. Jaco Ruit “Formal category theory in ∞\infty∞-equipments I” In arXiv preprint arXiv:2308.03583, 2023
  76. The sHoTT Community “sHoTT Library in Rzk”, 2024 URL: https://rzk-lang.github.io/sHoTT/
  77. Michael Shulman “All (∞,1)1(\infty,1)( ∞ , 1 )-toposes have strict univalent universes”, 2019 arXiv: https://arxiv.org/abs/1904.07004
  78. Michael Shulman “The univalence axiom for elegant Reedy presheaves” In Homology Homotopy Appl. 17.2, 2015, pp. 81–106 DOI: 10.4310/HHA.2015.v17.n2.a6
  79. Michael Shulman “Univalence for inverse diagrams and homotopy canonicity” In Math. Structures Comput. Sci. 25.5, 2015, pp. 1203–1277 DOI: 10.1017/S0960129514000565
  80. Michael Shulman “Univalence for inverse EI diagrams” In Homology Homotopy Appl. 19.2, 2017, pp. 219–249 DOI: 10.4310/HHA.2017.v19.n2.a12
  81. Mike Shulman “An explicit description of cocomma-categories?” version: 2016-08-11, MathOverflow URL: https://mathoverflow.net/q/247311
  82. Raffael Stenzel “Univalence and completeness of Segal objects” arXiv, 2019 DOI: 10.48550/ARXIV.1911.06640
  83. Danny Stevenson “Model structures for correspondences and bifibrations” In arXiv preprint arXiv:1807.08226, 2018
  84. Ross Street “Correction to: “Fibrations in bicategories” [Cahiers Topologie Géom. Différentielle 21 (1980), no. 2, 111–160; MR0574662 (81f:18028)]” In Cahiers Topologie Géom. Différentielle Catég. 28.1, 1987, pp. 53–56
  85. Ross Street “Elementary cosmoi. I”, Category Sem., Proc., Sydney 1972/1973, Lect. Notes Math. 420, 134-180 (1974)., 1974
  86. 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
  87. Ross Street “Fibrations in bicategories” In Cahiers Topologie Géom. Différentielle 21.2, 1980, pp. 111–160 URL: http://www.numdam.org/article/CTGDC_1980__21_2_111_0.pdf
  88. Thomas Streicher “A model of type theory in simplicial sets: a brief introduction to Voevodsky’s homotopy type theory” In J. Appl. Log. 12.1, 2014, pp. 45–49 DOI: 10.1016/j.jal.2013.04.001
  89. Thomas Streicher “Fibered Categories à la Jean Bénabou”, 2022 arXiv: https://arxiv.org/abs/1801.02927
  90. Andrew Swan “Separating Path and Identity Types in Presheaf Models of Univalent Type Theory”, 2018 arXiv: https://arxiv.org/abs/1808.00920
  91. The Univalent Foundations Program “Homotopy Type Theory: Univalent Foundations of Mathematics” Institute for Advanced Study: https://homotopytypetheory.org/book, 2013
  92. Vladimir Voevodsky “A simple type system with two identity types” Unpublished note. https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/HTS.pdf, 2013
  93. Vladimir Voevodsky “Notes on type systems” Unpublished, https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/expressions_current.pdf, 2009
  94. Tamara Von Glehn “Polynomials, fibrations and distributive laws” In Theory and Applications of Categories 33.36, 2018, pp. 1111–1144
  95. Michael Warren “Directed Type Theory” Lecture at IAS, Princeton, 2013 URL: https://www.youtube.com/watch?v=znn6xEZUKNE
  96. 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
  97. 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
  98. Jonathan Weinberger “Internal sums for synthetic fibered (∞,1)1(\infty,1)( ∞ , 1 )-categories”, 2022 arXiv:2205.00386 [math.CT]
  99. Jonathan Weinberger “Strict stability of extension types” arXiv, 2022 DOI: 10.48550/ARXIV.2203.07194
  100. Nobuo Yoneda “On Ext and exact sequences” In J. Fac. Sci. Univ. Tokyo Sect. I 8.507-576, 1960, pp. 1960
Citations (8)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now