We prove that the category of models of any relational Horn theory satisfying a mild syntactic condition is infinitely extensive. Central examples of such categories include the categories of preordered sets and partially ordered sets, and the categories of small V-categories, (symmetric) pseudo-V-metric spaces, and (symmetric) V-metric spaces for a commutative unital quantale V. We also explicitly characterize initial sources and final sinks in such categories, and in particular embeddings and quotients.
Keywords: relational Horn theory; extensive category; topological category; locally presentable category; concrete category; distributive category
2020 MSC: 06A06, 06F07, 18B50, 18C10, 18C35
Theory and Applications of Categories, Vol. 38, 2022, No. 23, pp 898-912.