On pure monomorphisms and pure epimorphisms in accessible categories

TL;DR

This paper investigates the properties of pure monomorphisms and epimorphisms in accessible categories, extending known results from additive to non-additive contexts under mild assumptions, and introduces Quillen exact classes.

Contribution

It generalizes the behavior of pure monomorphisms and epimorphisms to broader accessible categories with finite products or coproducts, and introduces Quillen exact classes.

Findings

Pure monomorphisms are directed colimits of split monomorphisms in categories with finite products.

Pure epimorphisms are directed colimits of split epimorphisms in categories with finite coproducts.

Introduction of Quillen exact classes generalizing one-sided exact categories.

Abstract

In all $\kappa$-accessible additive categories, $\kappa$-pure monomorphisms and $\kappa$-pure epimorphisms are well-behaved, as shown in our previous paper arXiv:2311.02418. This is known to be not always true in $\kappa$-accessible nonadditive categories. Nevertheless, mild assumptions on a $\kappa$-accessible category are sufficient to prove good properties of $\kappa$-pure monomorphisms and $\kappa$-pure epimorphisms. In particular, in a $\kappa$-accessible category with finite products, all $\kappa$-pure monomorphisms are $\kappa$-directed colimits of split monomorphisms, while in a $\kappa$-accessible category with finite coproducts, all $\kappa$-pure epimorphisms are $\kappa$-directed colimits of split epimorphisms. We also discuss what we call Quillen exact classes of monomorphisms and epimorphisms, generalizing the additive concept of one-sided exact category.