Hypergraph Categories

Understand

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".