---
tag: VT8SFWT0XW
kind: concept
name: Braided monoidal category
msc: [18M15]
specializes: [M3MSQ3NKAE]
sources:
  - url: https://en.wikipedia.org/wiki/Braided_monoidal_category
same_as:
  wikidata: Q2943034
  wikipedia: Braided monoidal category
---

## Summary

A braided monoidal category is a [[monoidal category|M3MSQ3NKAE]] with a natural isomorphism $A \otimes B \to B \otimes A$, the braiding, compatible with the associator.
The braiding need not square to the identity, and its behaviour is modelled by the [[braid groups|BNR4H5GCTP]] rather than the symmetric groups.
Braided monoidal categories arise from representations of quantum groups and give invariants of knots and links.
When the braiding squares to the identity the category is symmetric.

## Explanations

### L1 human

In a braided monoidal category two objects placed side by side can swap places.
Picture each object as a string: swapping them makes one string cross over the other.
Crossing over and crossing under are different moves, so swapping twice need not bring things back to the start.
For vector spaces the flip $v \otimes w \mapsto w \otimes v$ is a braiding, and there swapping twice does give the identity.