On extensivity of morphisms

TL;DR

This paper investigates the concept of extensivity in categories through morphisms, introducing extensive and coextensive morphisms, and connects categorical properties with universal algebra, establishing new equivalences in Barr-exact categories.

Contribution

It introduces the notions of extensive and coextensive morphisms, linking categorical properties with universal algebra concepts, and proves a new characterization of coextensivity in Barr-exact categories.

Findings

A Barr-exact category is coextensive iff every split monomorphism is coextensive.

The paper formalizes extensivity of morphisms and relates it to algebraic properties.

Universal algebra concepts are expressed categorically, enabling new generalizations.

Abstract

Extensivity of a category may be described as a property of coproducts in the category, namely, that they are disjoint and universal. An alternative viewpoint is that it is a property of morphisms in a category. This paper explores this point of view through a natural notion of extensive and coextensive morphism. Through these notions, topics in universal algebra, such as the strict refinement and Fraser-Horn properties, take categorical form and thereby enjoy the benefits of categorical generalisation. On the other hand, the universal algebraic theory surrounding these topics inspire categorical results. One such result we establish in this paper is that a Barr-exact category is coextensive if and only if every split monomorphism in the category is coextensive.