# Decorated corelations

## Brendan Fong

Let $C$ be a category with finite colimits, and let $(E, M)$ be a factorisation system on $C$ with $M$ stable under pushout. Writing $C;M^{\op}$ for the symmetric monoidal category with morphisms cospans of the form $\stackrel{c}\to \stackrel{m}\leftarrow$, where $c \in C$ and $m \in M$, we give a method for constructing a category from a symmetric lax monoidal functor $F : (C; \mc M^{\op},+) \to (Set,\times)$. A morphism in this category, termed a decorated corelation, comprises (i) a cospan $X \to N \leftarrow Y$ in $C$ such that the canonical copairing $X+Y \to N$ lies in $E$, 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 are useful for modelling their semantics.

Keywords: decorated cospan, corelation, Frobenius monoid, hypergraph category, well-supported compact closed category

2010 MSC: 18C10, 18D10

Theory and Applications of Categories, Vol. 33, 2018, No. 22, pp 608-643.

Published 2018-06-25.

TAC Home