On pure monomorphisms and pure epimorphisms in accessible categories
Leonid Positselski
TL;DR
This work extends κ‑purity from additive to broad nonadditive accessible settings by identifying mild, natural hypotheses under which κ‑pure monomorphisms and κ‑pure epimorphisms retain good structural behavior. It introduces strongly κ‑pure notions and two nonadditive generalizations, QE‑mono and QE‑epi classes (Quillen‑exact analogues), and develops a robust framework for constructing locally κ‑coherent and strongly locally κ‑coherent classes via directed limits of presentable data. Key results show that, under very weak cokernel/split pullback conditions, κ‑pure monos/epis decompose as directed colimits of split morphisms between κ‑presentables, and behave well with pushouts and pullbacks; in additive contexts, these recover classical exactness properties. The approach unifies and extends Purity theory beyond additivity, enabling nonadditive analogues of right/left exact categories and offering practical criteria (Plce‑style) for identifying QE‑monos/epos and their coherence properties.
Abstract
In all $κ$-accessible additive categories, $κ$-pure monomorphisms and $κ$-pure epimorphisms are well-behaved, as shown in our previous paper arXiv:2311.02418. This is known to be not always true in $κ$-accessible nonadditive categories. Nevertheless, mild assumptions on a $κ$-accessible category are sufficient to prove good properties of $κ$-pure monomorphisms and $κ$-pure epimorphisms. In particular, in a $κ$-accessible category with finite products, all $κ$-pure monomorphisms are $κ$-directed colimits of split monomorphisms, while in a $κ$-accessible category with finite coproducts, all $κ$-pure epimorphisms are $κ$-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.