  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". 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.
