A note on the Huq-commutativity of normal monomorphisms

TL;DR

This paper provides an alternative criterion for Huq-commutativity of Bourn-normal monomorphisms in unital categories, extending known results from protomodular contexts to more general settings.

Contribution

It introduces a new criterion for Huq-commutativity and generalizes several known facts from protomodular categories to broader unital categories.

Findings

Huq-commutativity characterized by trivial pullback in certain categories

Normal monomorphisms with same codomain Huq-commute under specific conditions

Extension of protomodular results to more general unital categories

Abstract

We give an alternative criteria for when a pair of Bourn-normal monomorphisms Huq-commute in a unital category. We use this to prove that in a unital category, in which a morphism is a monomorphism if and only if its kernel is zero morphism, a pair of Bourn-normal monomorphisms with the same codomain Huq-commute as soon as they have trivial pullback. As corollaries we show that several facts known only in the protomodular context are in fact true in more general contexts.