---
tag: A10WW8GEKR
kind: concept
name: Rigid monoidal category
msc: [18M10, 18M05]
specializes: [M3MSQ3NKAE]
sources:
  - url: https://en.wikipedia.org/wiki/Rigid_category
same_as:
  wikidata: Q7333804
  wikipedia: Rigid category
---

## Summary

A rigid monoidal category, also called an autonomous category, is a [[monoidal category|M3MSQ3NKAE]] in which every object has a left and a right [[dual object|8A7YXMW727]].
The main example is the category of finite-dimensional vector spaces, and more generally of finite-dimensional representations of a group.
Rigidity gives every morphism a transpose, and in the braided case it allows traces and dimensions.
Fusion categories and Tannakian categories are rigid by definition.

## Explanations

### L1 human

Every finite-dimensional vector space has a dual space, and every linear map has a transpose between the duals.
A rigid monoidal category is one where every object has such a dual.
In string diagrams, this means that every string can be bent up or down, turning inputs into outputs.
Infinite-dimensional vector spaces spoil this, so the category of all vector spaces is not rigid.