Effective descent morphisms of regular epimorphisms

TL;DR

This paper characterizes when regular epimorphisms in certain regular categories are effective descent morphisms, showing it depends on the regularity of the category of regular epimorphisms, with applications to various algebraic and topological categories.

Contribution

It establishes a necessary and sufficient condition for regular epimorphisms to be effective descent morphisms in regular categories, linking this property to the regularity of the category of regular epimorphisms.

Findings

Regular epimorphisms are effective descent morphisms iff $Reg(A)$ is regular.

In exact Goursat, ideal determined, or topological Mal'tsev categories, all regular epimorphisms are effective descent.

The result applies to categories of $n$-fold regular epimorphisms in these contexts.

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 $n\geq 1$.