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 $n\geq 1$.