Extensivity of a category [3] may be described as a property of coproducts - namely, that they are disjoint and universal. An alternative viewpoint is that it is a property of morphisms. 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 a categorical form and thereby enjoy the benefits of categorical generalisation. On the other hand, the universal algebraic theory surrounding these topics leads to categorical results. One such result we establish in this paper is that a Barr-exact category C is coextensive if and only if every split monomorphism in C is coextensive.
Keywords: Extensive categories, extensive morphisms, coextensive categories, strict refinement property, Fraser-Horn property
2020 MSC: 18B50, 18A20, 18A30, 08B25
Theory and Applications of Categories, Vol. 44, 2025, No. 21, pp 617-642.
Published 2025-07-14.
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