Effective descent morphisms of regular epimorphisms

Abstract

Let A be a regular category with pushouts of regular epimorphisms by regular epimorphism and Reg(A) the category of regular epimorphisms in A. We prove that every regular epimorphism in Reg(A) is an effective descent morphism if, and only if, Reg(A) is a regular category. Then, moreover, every regular epimorphism in A is an effective descent morphism. This is the case, for instance, when A is either exact Goursat, or ideal determined, or is a category of topological Mal’tsev algebras, or is the category of n-fold regular epimorphisms in any of the three previous cases, for any n≥1.