In all κ-accessible additive categories, κ-pure monomorphisms and κ-pure epimorphisms are well-behaved, as shown in our previous paper [L. Positselski, "Locally coherent exact categories", Appl. Categorical Struct. 32, 2024]. 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.
Keywords: κ-directed colimits, accessible categories, split monomorphisms, split epimorphisms, pure monomorphisms, pure epimorphisms, regular monomorphisms, regular epimorphisms, pushouts, pullbacks, finite products, finite coproducts, very weak cokernel pairs, very weak split pullbacks, one-sided Quillen exact categories, locally coherent Quillen exact classes
2020 MSC: 18C35, 18A20, 18A30, 18E05, 18E08
Theory and Applications of Categories, Vol. 45, 2026, No. 7, pp 246-288.
Published 2026-01-30.
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