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Generalizing quasi-categories via model structures on simplicial sets

Matthew Feller

Abstract

We use Cisinski's machinery to construct and study model structures on the category of simplicial sets whose classes of fibrant objects generalize quasi-categories. We identify a lifting condition which captures the homotopical behavior of quasi-categories without the algebraic aspects and show that there is a model structure whose fibrant objects are precisely those which satisfy this condition. We also identify a localization of this model structure whose fibrant objects satisfy a "special horn lifting" condition similar to the one satisfied by quasi-categories. This special horn model structure leads to a conjecture 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.

Generalizing quasi-categories via model structures on simplicial sets

Abstract

We use Cisinski's machinery to construct and study model structures on the category of simplicial sets whose classes of fibrant objects generalize quasi-categories. We identify a lifting condition which captures the homotopical behavior of quasi-categories without the algebraic aspects and show that there is a model structure whose fibrant objects are precisely those which satisfy this condition. We also identify a localization of this model structure whose fibrant objects satisfy a "special horn lifting" condition similar to the one satisfied by quasi-categories. This special horn model structure leads to a conjecture 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.

Paper Structure

This paper contains 23 sections, 43 theorems, 25 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Model structures on simplicial sets
  3. 2-Segal motivation
  4. Cisinski's theory and the minimal model structure
  5. Homotopically-behaved model structures and augmented horns
  6. Special horns
  7. Pointwise cylinders
  8. Organization
  9. Acknowledgement
  10. Background
  11. Basics of simplicial sets and model structures
  12. Cisinski's theory
  13. Homotopically-behaved model structures and augmented horns
  14. Augmented horn extensions
  15. Augmented triangulations
  16. ...and 8 more sections

Key Result

Theorem 1

There exists a minimal homotopically-behaved model structure on $\operatorname{sSet}$. The fibrant objects in this model structure are the simplicial sets with lifts of certain modified horn inclusions, which we call $J$-augmented horn inclusions.

Figures (11)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5:
  • ...and 6 more figures

Theorems & Definitions (134)

  • Theorem
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Example 2.6
  • Definition 2.7
  • Lemma 2.8
  • proof
  • ...and 124 more