Generalizing quasicategories via model structures on simplicial sets

Feller, Matt¹

¹Department of Mathematics, University of Virginia, Charlottesville, VA, United States

Algebraic and Geometric Topology, Tome 25 (2025) no. 1, pp. 357-397

Résumé

We use Cisinski’s machinery to construct and study model structures on the category of simplicial sets whose classes of fibrant objects generalize quasicategories. We identify a lifting condition that captures the homotopical behavior of quasicategories without the algebraic aspects and show that there is a model structure whose fibrant objects are precisely those that satisfy this condition. We also identify a localization of this model structure whose fibrant objects satisfy a “special horn lifting” property similar to the one satisfied by quasicategories. This special horn model structure leads to a conjectural characterization of the bijective-on-0-simplices trivial cofibrations of the Joyal model structure. We also discuss how these model structures all relate to one another and to the minimal model structure.

Détail

DOI : 10.2140/agt.2025.25.357

Keywords: simplicial sets, model categories, model structure

Bibliographie

[1] A Campbell, A counterexample in quasi-category theory, Proc. Amer. Math. Soc. 148 (2020) 37 | DOI

[2] A Campbell, E Lanari, On truncated quasi-categories, Cah. Topol. Géom. Différ. Catég. 61 (2020) 154 | DOI

[3] D C Cisinski, Les préfaisceaux comme modèles des types d’homotopie, 308, Soc. Math. France (2006)

[4] D C Cisinski, Higher categories and homotopical algebra, 180, Cambridge Univ. Press (2019) | DOI

[5] T Dyckerhoff, M Kapranov, Higher Segal spaces, 2244, Springer (2019) | DOI

[6] M Feller, A horn-like characterization of the fibrant objects in the minimal model structure on simplicial sets, High. Struct. 7 (2023) 166 | DOI

[7] M Feller, Quasi-2-Segal sets, Tunis. J. Math. 5 (2023) 327 | DOI

[8] I Gálvez-Carrillo, J Kock, A Tonks, Decomposition spaces, incidence algebras and Möbius inversion, I : Basic theory, Adv. Math. 331 (2018) 952 | DOI

[9] P G Goerss, J F Jardine, Simplicial homotopy theory, 174, Birkhäuser (1999) | DOI

[10] P S Hirschhorn, Model categories and their localizations, 99, Amer. Math. Soc. (2003) | DOI

[11] M Hovey, Model categories, 63, Amer. Math. Soc. (1999)

[12] A Joyal, Quasi-categories and Kan complexes, J. Pure Appl. Algebra 175 (2002) 207 | DOI

[13] A Joyal, Notes on quasi-categories, preprint (2008)

[14] A Joyal, The theory of quasi-categories and its applications, II, lecture notes (2008)

[15] D G Quillen, Homotopical algebra, 43, Springer (1967) | DOI

[16] C Rezk, Introduction to quasicategories, lecture notes (2022)