Compatible weak factorization systems and model structures

TL;DR

This paper introduces compatible weak factorization systems in general categories, providing a method to construct model structures that generalize known results and include classical examples like the Kan-Quillen model structure.

Contribution

It generalizes Gillespie's result by establishing a construction of model structures from compatible weak factorization systems in arbitrary categories.

Findings

Compatible weak factorization systems can be used to build model structures.

Classical model structures like Kan-Quillen satisfy the compatibility conditions.

The method extends the construction of model structures beyond abelian categories.

Abstract

In this paper the concept of compatible weak factorization systems in general categories is introduced as a counterpart of compatible complete cotorsion pairs in abelian categories. We describe a method to construct model structures on general categories via two compatible weak factorization systems satisfying certain conditions, and hence generalize a very useful result by Gillespie for abelian model structures. As particular examples, we show that weak factorizations systems associated to some classical model structures (for example, the Kan-Quillen model structure on $\mathsf{sSet}$) satisfy these conditions.