Enriched cofibration categories

TL;DR

This paper extends the concept of cofibration categories to enriched settings, showing that categories of enriched diagrams inherit cofibration structures under certain conditions, thus advancing homotopy theory frameworks.

Contribution

It introduces an enriched version of cofibration categories and proves that categories of enriched diagrams inherit cofibration structures under local cofibrancy or flatness.

Findings

Enriched cofibration categories are formalized.

Categories of enriched diagrams inherit cofibration structures.

Main result applies when the base category is locally cofibrant or flat.

Abstract

Cofibration categories are a formalization of homotopy theory useful for dealing with homotopy colimits that exist on the level of models as colimits of cofibrant diagrams. In this paper, we deal with their enriched version. Our main result claims that the category $[\mathcal{C},\mathcal{M}]$ of enriched diagrams equipped with the projective structure inherits a structure of a cofibration category whenever $\mathcal{C}$ is locally cofibrant (or, more generally, locally flat).