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Dwyer-Kan homotopy theory of enriched categories

Fernando Muro

Abstract

We construct a model structure on the category of small categories enriched over a combinatorial closed symmetric monoidal model category satisfying the monoid axiom. Weak equivalences are Dwyer-Kan equivalences, i.e. enriched functors which induce weak equivalences on morphism objects and equivalences of ordinary categories when we take sets of connected components on morphism objects.

Dwyer-Kan homotopy theory of enriched categories

Abstract

We construct a model structure on the category of small categories enriched over a combinatorial closed symmetric monoidal model category satisfying the monoid axiom. Weak equivalences are Dwyer-Kan equivalences, i.e. enriched functors which induce weak equivalences on morphism objects and equivalences of ordinary categories when we take sets of connected components on morphism objects.

Paper Structure

This paper contains 12 sections, 47 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. The bifibered categories of enriched categories and graphs
  3. Homotopy theory of enriched categories with fixed set of objects
  4. Generating (trivial) cofibrations
  5. Push-outs of free enriched functors
  6. Pseudo-cofibrations
  7. Endomorphism monoids of cofibrant enriched categories
  8. The monoid axiom
  9. Closure properties of homotopically essentially surjective enriched functors
  10. The Dwyer--Kan model structure
  11. Change of base category
  12. A kind of left properness result

Key Result

Theorem 1.1

Let $\mathscr{V}$ be a combinatorial closed symmetric monoidal model category satisfying Schwede--Shipley's monoid axiom ammmc. Then $\operatorname{Cat}_{}(\mathscr{V})$ admits the Dwyer--Kan model structure. Moreover, this model structure is combinatorial.

Theorems & Definitions (87)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.6
  • ...and 77 more