# Hypergraph Categories

arXiv: https://arxiv.org/abs/1806.08304 · HTML: https://mlsanity.com/paper/1806.08304/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jun 2018.

## Built on by (most cited first, 7 of 7)

- [On the Anatomy of Attention](https://mlsanity.com/paper/2407.02423/) (2024)
- [An Algebraic Approach to Moralisation and Triangulation of Probabilistic Graphical Models](https://mlsanity.com/paper/2503.11820/) (2025)
- [Generalized Gradient Descent is a Hypergraph Functor](https://mlsanity.com/paper/2403.19845/) (2024)
- [Aggregating time-series and image data: functors and double functors](https://mlsanity.com/paper/2504.05274/) (2025)
- [Bayesian Networks, Markov Networks, Moralisation, Triangulation: a Categorical Perspective](https://mlsanity.com/paper/2512.09908/) (2025)
- [Categorical Belief Propagation: Sheaf-Theoretic Inference via Descent and Holonomy](https://mlsanity.com/paper/2601.04456/) (2026)
- [Global Coherence: When Every Agent Is Right and the Team Is Still Wrong - A Local-to-Global Semantic Foundation for Multi-Agent Collaboration](https://mlsanity.com/paper/2610.02036/) (2026)

## Abstract (arXiv)

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.