Hypergraph categories

Abstract

Hypergraph categories have been rediscovered at least five times, under various names, including well-supported compact closed categories, dgs-monoidal categories, and dungeon categories. Perhaps the reason they keep being reinvented is two-fold: there are many applications—including to automata, databases, circuits, linear relations, graph rewriting, and belief propagation—and yet the standard definition is so involved and ornate as to be difficult to find in the literature. Indeed, a hypergraph category is, roughly speaking, a “symmetric monoidal category in which each object is equipped with the structure of a special commutative Frobenius monoid, satisfying certain coherence conditions”.Fortunately, this description can be simplified a great deal: a hypergraph category is simply a “cospan-algebra,” roughly a lax monoidal functor from cospans to sets. The goal of this paper is to remove the scare-quotes and make the previous statement precise. We prove two main theorems. First is a coherence theorem for hypergraph categories, which says that every hypergraph category is equivalent to an objectwise-free hypergraph category. Second, we prove that the category of objectwise-free hypergraph categories is equivalent to the category of cospan-algebras.