On extensivity of morphisms
Michael Hoefnagel, Emma Theart
TL;DR
This work reframes extensivity as a property of morphisms, introducing extensive and coextensive morphisms and deriving a rich interplay between universal algebra and category theory. It develops core criteria for (co)extensivity via (E1)/(E2) and (C1)/(C2), explores the impact of coproduct inclusions and product projections, and connects these notions to the strict refinement and Fraser–Horn properties. A central result shows that in Barr-exact categories, coextensivity is equivalent to every split monomorphism being coextensive, with further implications for regular epimorphisms and pullback/pushout behavior. The paper also provides concrete examples (posets, semilattices, centerless monoids, UD-categories) and extends the framework to M-extensivity, establishing a broad categorical paradigm for extending universal-algebraic ideas to morphisms and objects alike.
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.