Pursuing Lax Diagrams and Enrichment

TL;DR

This paper develops a homotopy-theoretic framework for enriched categories using lax diagrams and co-Segal categories, removing strong assumptions to establish a localized model structure where co-Segal and strict categories are equivalent.

Contribution

It introduces unital co-Segal M-categories, removes previous strong hypotheses, and constructs a Bousfield localization for a non-left proper model category.

Findings

Established a model structure where co-Segal M-categories are equivalent to strict M-categories.

Removed strong assumptions previously needed for unital co-Segal categories.

Provided foundational constructions for future applications in enriched category theory.

Abstract

This paper is part of a project that aims to give a homotopy cousin of Kelly's treatment of enriched category theory. After introducing unital co-Segal M-categories, we establish the unital version of a previous theorem that was proven for the nonunital ones; but this was done under strong hypothesis. We've removed here these assumptions and try to keep the hypothesis on M as minimal as possible. Our main result provides a sort of Bousfield localization of a model category that is not known to be left proper. In this model structure, every co-Segal M-category is `canonically' equivalent to a strict M-category with the same set of objects. We revisit some constructions of classical enriched category theory to set up the necessary material for our future applications.