Effective descent morphisms of regular epimorphisms
This paper is concerned with a weakened form of Barr exactness: the condition that every regular epimorphism in a category should be an effective descent morphism. In particular, let \({\mathcal A}\) be a regular category with pushouts of regular epimorphisms by regular epimorphisms, and let \(\text{Reg}({\mathcal A})\) be the category of regular epimorphisms in \({\mathcal A}\). The author shows that \(\text{Reg}({\mathcal A})\) satisfies the weakened exactness condition if and only if \(\text{Reg}({\mathcal A})\) is regular. When these conditions hold, the category \({\mathcal A}\) itself also satisfies the weakened exactness condition. All of the conditions are satisfied when \({\mathcal A}\) is a Goursat category, or an ideal determined category, or a category of topological Mal'tsev algebras. They are also satisfied when \({\mathcal A}\) is the category of \(n\)-fold regular epimorphisms in one of the categories just mentioned.
scientific article; zbMATH DE number 6094150