Model categories with simple homotopy categories

TL;DR

This paper develops methods to construct model structures on bicomplete categories, focusing on when a subcategory can serve as weak equivalences, with applications including infinite graphs.

Contribution

It introduces new constructions of model structures on bicomplete categories based on subcategories of weak equivalences, especially via functors to other categories.

Findings

Properness of the constructions under certain conditions

Application to the category of infinite graphs

Conditions for subcategories to serve as weak equivalences

Abstract

In the present article, we describe constructions of model structures on general bicomplete categories. We are motivated by the following question: given a category $\mathcal{C}$ with a subcategory $w\mathcal{C}$ closed under retracts, when is there a model structure on $\mathcal{C}$ with $w\mathcal{C}$ as the subcategory of weak equivalences? We begin exploring this question in the case where $w\mathcal{C} = F^{-1}(\mathrm{iso}\, \mathcal{D})$ for some functor $F:\mathcal{C}\rightarrow \mathcal{D}$. We also prove properness of our constructions under minor assumptions and examine an application to the category of infinite graphs.