Hypergraph categories

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:223

Article

关键词: Hypergraph categories;    Compact closed categories;    Frobenius algebras;    Cospan;    Wiring diagram;   

DOI  :  10.1016/j.jpaa.2019.02.014

来源: Elsevier

【 摘 要 】

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 twofold: 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. (C) 2019 Elsevier B.V. All rights reserved.

【 授权许可】

Free