On extensivity of morphisms

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.