Homotopy locally presentable enriched categories

TL;DR

This paper develops a homotopy theory for categories enriched in a monoidal model category, extending known results to a broader context involving homotopy limits, colimits, and local presentability.

Contribution

It generalizes the concept of homotopy local presentability to enriched categories over a monoidal model category, connecting it with combinatorial model categories.

Findings

Established a link between homotopy locally presentable V-categories and combinatorial model V-categories.

Extended the theory from simplicially-enriched categories to more general monoidal model categories.

Provided a framework for homotopy weighted limits and colimits in enriched categories.

Abstract

We develop a homotopy theory of categories enriched in a monoidal model category V. In particular, we deal with homotopy weighted limits and colimits, and homotopy local presentability. The main result, which was known for simplicially-enriched categories, links homotopy locally presentable V-categories with combinatorial model V-categories, in the case where has all objects of V are cofibrant.