Generalizing quasicategories via model structures on simplicial sets

Feller, Matt

doi:10.2140/agt.2025.25.357

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Generalizing quasicategories via model structures on simplicial sets

Matt Feller

Algebraic & Geometric Topology 25 (2025) 357–397

DOI: 10.2140/agt.2025.25.357

Abstract

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.

Keywords

simplicial sets, model categories, model structure

Mathematical Subject Classification

Primary: 18N50, 55U35

Secondary: 18N60

References

Publication

Received: 23 November 2021

Revised: 22 May 2023

Accepted: 30 September 2023

Published: 24 March 2025

Authors

Matt Feller
Department of Mathematics University of Virginia Charlottesville, VA United States

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Volume 25, issue 1 (2025)