---
tag: D7NA6T33A7
kind: concept
name: Fusion category
msc: [18M20, 18M10]
specializes: [A10WW8GEKR]
sources:
  - url: https://en.wikipedia.org/wiki/Fusion_category
same_as:
  wikidata: Q30672665
  wikipedia: Fusion category
---

## Summary

A fusion category is a semisimple [[rigid monoidal category|A10WW8GEKR]] over a field, with finite-dimensional spaces of morphisms.
It has finitely many simple objects up to isomorphism, and its unit object is simple.
Its structure is recorded by fusion rules: how the tensor product of two simple objects splits into simple objects.
The finite-dimensional representations of a finite group over $\C$ form the basic example.
Fusion categories appear in subfactor theory, in conformal field theory and in topological phases of matter.

## Explanations

### L1 human

A fusion category is a finite "multiplication table" for building blocks that can be combined and decomposed.
There are finitely many basic objects, every object is a sum of basic ones, and combining two basic objects gives a sum of basic objects again.
For example, the group $\Z/2$ has two irreducible complex representations, the trivial one $1$ and the sign $s$.
Their tensor products are $1 \otimes 1 = 1$, $1 \otimes s = s$ and $s \otimes s = 1$.