When a matrix condition implies the Mal'tsev property

TL;DR

This paper investigates when matrix conditions, which extend algebraic properties to category theory, imply the Mal'tsev property, showing that this implication is equivalent to the case within varieties of universal algebras.

Contribution

It characterizes the conditions under which matrix conditions imply the Mal'tsev property in both finitely complete and regular categories, linking it to varieties of universal algebras.

Findings

Implication holds if and only if it holds in varieties of universal algebras.

Results unify conditions for finitely complete and regular categories.

Provides criteria for when matrix conditions imply Mal'tsev property.

Abstract

Matrix conditions extend linear Mal'tsev conditions from Universal Algebra to exactness properties in Category Theory. Some can be stated in the finitely complete context while, in general, they can only be stated for regular categories. We study when such a matrix condition implies the Mal'tsev property. Our main results assert that, for both types of matrices, this implication is equivalent to the corresponding implication restricted to the context of varieties of universal algebras.