Exponentiability in categories of relational structures

For a relational Horn theory $\mathbb{T}$, we provide useful sufficient conditions for the exponentiability of objects and morphisms in the category $\mathbb{T}\text{-}\mathsf{Mod}$ of $\mathbb{T}$-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 $\mathbb{T}\text{-}\mathsf{Mod}$ 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 $V\text{-}\mathsf{Cat}$ of small $V$-categories for certain commutative unital quantales $V$.

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