Hypergraph Categories

06/21/2018
by   Brendan Fong, et al.
0

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. In doing so, we identify a new unit coherence axiom, which seems to have been overlooked in the literature, but which is necessary for the hypergraph strictification result and hence we contend should be included in the definition. Second, we prove that the category of objectwise-free hypergraph categories is equivalent to the category of cospan-algebras.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

12/27/2017

Rewriting in Free Hypergraph Categories

We study rewriting for equational theories in the context of symmetric m...
12/27/2017

Rewriting in Free Hypegraph Categories

We study rewriting for equational theories in the context of symmetric m...
06/23/2019

Reflecting Algebraically Compact Functors

A compact T-algebra is an initial T-algebra whose inverse is a final T-c...
10/11/2019

Categories for Me, and You?

A non-self-contained gathering of notes on category theory, including th...
07/30/2018

Coherence for braided distributivity

In category-theoretic models for the anyon systems proposed for topologi...
01/11/2021

Deductive Systems and Coherence for Skew Prounital Closed Categories

In this paper, we develop the proof theory of skew prounital closed cate...
04/17/2020

*-autonomous envelopes and 2-conservativity of duals

We show the doctrine of ∗-autonomous categories is "2-conservative" over...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.