On model structure for coreflective subcategories of a model category

TL;DR

This paper demonstrates that certain coreflective subcategories of cofibrantly generated model categories can themselves be equipped with compatible model structures, with applications to categories of topological spaces.

Contribution

It establishes conditions under which coreflective subcategories inherit cofibrantly generated model structures that are Quillen equivalent to the parent category.

Findings

Coreflective subcategories can admit cofibrantly generated model structures.

Well-known categories of topological spaces have finitely generated model structures.

These model structures are Quillen equivalent to the standard topological space model.

Abstract

Let $\bf C$ be a coreflective subcategory of a cofibrantly generated model category $\bf D$. In this paper we show that under suitable conditions $\bf C$ admits a cofibrantly generated model structure which is left Quillen adjunct to the model structure on $\bf D$. As an application, we prove that well-known convenient categories of topological spaces, such as $k$-spaces, compactly generated spaces, and $\Delta$-generated spaces \cite{DN} (called numerically generated in \cite{KKH}) admit a finitely generated model structure which is Quillen equivalent to the standard model structure on the category $\bf Top$ of topological spaces.