Beck-Chevalley condition and Goursat categories

TL;DR

This paper characterizes regular Goursat categories using a stability property of regular epimorphisms and a restricted Beck-Chevalley condition, providing structural insights into 3-permutable varieties of universal algebras.

Contribution

It introduces a new stability property and a restricted Beck-Chevalley condition to characterize Goursat categories and explains the algebraic structure of 3-permutable varieties.

Findings

Characterization of Goursat categories via stability of regular epimorphisms.

Expression of this property as a restricted Beck-Chevalley condition.

Structural explanation for ternary operations in 3-permutable varieties.

Abstract

We characterise regular Goursat categories through a specific stability property of regular epimorphisms with respect to pullbacks. Under the assumption of the existence of some pushouts this property can be also expressed as a restricted Beck-Chevalley condition, with respect to the fibration of points, for a special class of commutative squares. In the case of varieties of universal algebras these results give, in particular, a structural explanation of the existence of the ternary operations characterising $3$-permutable varieties of universal algebras.