Pursuing Lax Diagrams and Enrichment

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.

70 pages, first draft, comments always appreciated. arXiv admin note: text overlap with arXiv:math/0608040 by other authors

References in corpus (5)

• Enriched $\infty$-categories via non-symmetric $\infty$-operads
• Weak enriched categories - Categories enrichies faibles
• Segal Enriched Categories I
• Lax Diagrams and Enrichment
• Les groupements

Cited by in corpus (1)

• Left Bousfield localization without left properness