Beck-Chevalley condition and Goursat categories

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.

Revised version: the proof of Theorem 1.3 has been changed. The paper is going to be published in the Journal of Pure and Applied Algebra