The Category Rel(Nom)

Abstract

The category ${\rm Rel}(\mathcal{C})$ may be formed for any category $\mathcal{C}$ with finite limits using the same objects as $\mathcal{C}$ but whose morphisms from $X$ to $Y$ are binary relations in $\mathcal{C}$, that is, subobjects of $X\times Y$. In this paper, concerning the topos ${\bf Nom}$, we study the category ${\bf Rel}({\bf Nom})$. In this category, we define and investigate certain morphisms, such as deterministic morphisms. Then, stochastic mappings between nominal sets are defined by exploiting the underlying relation of functions between nominal sets. This allows one to reinterpret concepts and earlier results in terms of morphisms.