Decorated Corelations

Let $\mathcal{C}$ be a category with finite colimits, and let $(\mathcal E,\mathcal M)$beafactorisationsystemon$\mathcal C$with$\mathcal M$stableunderpushouts\mathrm{.}Writing$\mathcal C;\mathcal M^{$\mathrm{o}\mathrm{p}$}$ for the symmetric monoidal category with morphisms cospans of the form $\stackrel{c}\to \stackrel{m}\leftarrow$,where$c \in \mathcal C$and$m \in \mathcal M$, we give method for constructing a category from a symmetric lax monoidal functor $F\text{\hspace{0.17em}⁣}:(\mathcal{C};{\mathcal{M}}^{\mathrm{o}\mathrm{p}},+)\to (\mathrm{S}\mathrm{e}\mathrm{t},\times )$. A morphism in this category, termed a \emph{decorated corelation}, comprises (i) a cospan $X\to N\leftarrow Y$ in $\mathcal{C}$ such that the canonical copairing $X+Y\to N$ lies in $\mathcal{E}$, together with (ii) an element of $FN$. Functors between decorated corelation categories can be constructed from natural transformations between the decorating functors $F$. This provides a general method for constructing hypergraph categories—symmetric monoidal categories in which each object is a special commutative Frobenius monoid in a coherent way—and their functors. Such categories are useful for modelling network languages, for example circuit diagrams, and such functors their semantics.