Decorated corelations

Abstract

Let C be a category with finite colimits, and let (ε, M) be a factorisation system on C with M stable under pushout. Writing C; Mop for the symmetric monoidal category with morphisms cospans of the form cm→←, where c ∈ C and m ∈ M, we give a method for constructing a category from a symmetric lax monoidal functor F: (C; Mop, +) → (Set, ×). A morphism in this category, termed a decorated corelation, comprises (i) a cospan X → N ← Y in C such that the canonical copairing X + Y → N lies in ε, together with (ii) an element of F N. 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.