The Frobenius equivalence and Beck-Chevalley condition for Algebraic Weak Factorisation Systems
Abstract: If a locally cartesian closed category carries a weak factorisation system, then the left maps are stable under pullback along right maps if and only if the right maps are closed under pushforward along right maps. We refer to this statement as the Frobenius equivalence and in this paper we state and prove an analogical statement for algebraic weak factorisation systems. These algebraic weak factorisation systems are an explicit variant of the more traditional weak factorisation systems in that the factorisation and the lifts are part of the structure of an algebraic weak factorisation system and are not merely required to exist. Our work has been motivated by the categorical semantics of type theory, where the Frobenius equivalence provides a useful tool for constructing dependent function types. We illustrate our ideas using split fibrations of groupoids, which are the backbone of the groupoid model of Hofmann and Streicher.
- Steve Awodey and Michael A. Warren “Homotopy theoretic models of identity types” In Mathematical Proceedings of the Cambridge Philosophical Society 146.1, 2009, pp. 45–55 DOI: 10.1017/S0305004108001783
- John Bourke “An orthogonal approach to algebraic weak factorisation systems” In Journal of Pure and Applied Algebra 227.6, 2023, pp. 107294 DOI: 10.1016/j.jpaa.2022.107294
- “Algebraic weak factorisation systems I: Accessible AWFS” In Journal of Pure and Applied Algebra 220.1, 2016, pp. 108–147 DOI: 10.1016/j.jpaa.2015.06.002
- P. J. Freyd and G. M. Kelly “Categories of continuous functors, I” In Journal of Pure and Applied Algebra 2.3, 1972, pp. 169–191 DOI: 10.1016/0022-4049(72)90001-1
- Nicola Gambino and Marco Federico Larrea “Models of Martin-Löf type theory from algebraic weak factorisation systems” In The Journal of Symbolic Logic 88.1, 2023, pp. 242–289 DOI: 10.1017/jsl.2021.39
- “The Frobenius condition, right properness, and uniform fibrations” In Journal of Pure and Applied Algebra 221.12, 2017, pp. 3027–3068 DOI: 10.1016/j.jpaa.2017.02.013
- Richard Garner “Understanding the small object argument” In Applied Categorical Structures 17.3, 2009, pp. 247–285 DOI: 10.1007/s10485-008-9137-4
- “Natural weak factorization systems” In Archivum Mathematicum 42.4 Department of Mathematics, Faculty of Science of Masaryk University, Brno, 2006, pp. 397–408
- “A 2-categorical proof of Frobenius for fibrations defined from a generic point” arXiv, 2024 arXiv:2210.00078 [math]
- Philip S. Hirschhorn “Model categories and their localizations” 99, Mathematical Surveys and Monographs American Mathematical Society, 2003 DOI: 10.1090/surv/099
- “The groupoid interpretation of type theory” In Twenty Five Years of Constructive Type Theory Oxford University Press, 1998, pp. 83–111 DOI: 10.1093/oso/9780198501275.003.0008
- Andre Joyal “Notes on clans and tribes” arXiv, 2017 arXiv:1710.10238 [math]
- 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 DOI: 10.4171/jems/1050
- Marco Federico Larrea “Models of dependent type theory from algebraic weak factorisation systems”, 2018
- Paige Randall North “Type theoretic weak factorization systems”, 2017 DOI: 10.17863/CAM.11207
- The Univalent Foundations Program “Homotopy type theory: Univalent foundations of mathematics”, 2023 DOI: 10.48550/arXiv.1308.0729
- Benno van den Berg and Eric Faber “Effective trivial Kan fibrations in simplicial sets” Cham: Springer International Publishing, 2022 DOI: 10.1007/978-3-031-18900-5˙8
- Benno van den Berg and Richard Garner “Topological and simplicial models of identity types” In ACM Transactions on Computational Logic 13.1, 2012, pp. 3:1–3:44 DOI: 10.1145/2071368.2071371
- Benno van den Berg and Freek Geerligs “Examples and cofibrant generation of effective Kan fibrations” arXiv, 2024 arXiv:2402.10568 [math]
- Wijnand Koen van Woerkom “Algebraic models of type theory”, 2021
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.