For a relational Horn theory T, we provide useful sufficient conditions for the exponentiability of objects and morphisms in the category T-mod of T-models; well-known examples of such categories, which have found recent applications in the study of programming language semantics, include the categories of preordered sets and (extended) metric spaces. As a consequence, we obtain useful sufficient conditions for T-mod to be cartesian closed, locally cartesian closed, and even a quasitopos; in particular, we provide two different explanations for the cartesian closure of the categories of preordered and partially ordered sets. Our results recover (the sufficiency of) certain conditions that have been shown by Niefield and Clementino-Hofmann to characterize exponentiability in the category of partially ordered sets and the category V-cat of small V-categories for certain commutative unital quantales V.
Keywords: relational Horn theory; relational structure; exponentiability; cartesian closed; locally cartesian closed; partial product; quasitopos
2020 MSC: 06A06, 06F07, 08A02, 18C35, 18D15
Theory and Applications of Categories, Vol. 39, 2023, No. 16, pp 493-518.