Surjection-like classes of morphisms

Pierre-Alain Jacqmin

We characterize "good" classes of epimorphisms in a finitely complete category, i.e., those which "interact with finite limits as surjections do in the category Set of sets and functions". More precisely, we prove that given a class E of morphisms in a small finitely complete category C, there exists a faithful conservative (respectively fully faithful) embedding C -> Set^D into a presheaf category which preserves and reflects finite limits and which sends morphisms in E, and only those, to componentwise surjections if and only if E contains the identities, is closed under composition, has the strong right cancellation property, is stable under pullbacks and does not contain any proper monomorphisms (respectively any morphism in it is a regular epimorphism). The classes of split epimorphisms and descent morphisms are such examples and the corresponding full embedding theorems are given by Yoneda and Barr's embeddings. As new examples, we get a conservative embedding theorem for the class of pullback-stable strong epimorphisms and a full embedding theorem for the class of effective descent morphisms. The proof presented here is not based on transfinite inductions and is therefore rather explicit, in contrast with similar embedding theorems.

Keywords: class of epimorphisms, embedding theorem, finite limit, pullback-stable strong epimorphism, descent morphism, effective descent morphism, split epimorphism, finite limit preserving functor

2020 MSC: 18A20, 18B15, 18A30 (primary); 18A35, 18A25, 18A22, 18G05 (secondary)

Theory and Applications of Categories, Vol. 39, 2023, No. 32, pp 949-1013.

Published 2023-12-04.

TAC Home