On the homotopy theory of enriched categories

TL;DR

This paper establishes conditions for creating Quillen model structures on small categories enriched over a monoidal model category, unifying various known structures like simplicial, topological, dg-, and spectral categories.

Contribution

It introduces the Interval Cofibrancy Theorem and provides a unified framework for model structures on enriched categories.

Findings

Unified treatment of known enriched category model structures

Sufficient conditions for model structure existence

Fundamental role of the Interval Cofibrancy Theorem

Abstract

We give sufficient conditions for the existence of a Quillen model structure on small categories enriched in a given monoidal model category. This yields a unified treatment for the known model structures on simplicial, topological, dg- and spectral categories. Our proof is mainly based on a fundamental property of cofibrant enriched categories on two objects, stated below as the Interval Cofibrancy Theorem.