# nLab ribbon category

Contents

### Context

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# Contents

## Idea

A ribbon category [Reshetikhin & Turaev (1990)] (also called a tortile category [Joyal & Street (1993)] or balanced rigid braided tensor category) is a monoidal category $(\mathcal{C}, \otimes, \mathbb{1}, \alpha, l, r)$ equipped with braiding $\beta=\{\beta_{X,Y}\}$, twist $\theta=\{\theta_X\}$ and duality $(\vee, b, d)$ that satisfy some compatibility conditions.

## Definition

A braided monoidal category is a monoidal category $\mathcal{C}$ equipped with a braiding $\beta$, which is a natural isomorphism $\beta_{X,Y}\colon X \otimes Y \to Y \otimes X$ obeying the hexagon identities.

A braided monoidal category is rigid if, for every object $X$, there exist objects $X^{\vee}$ and ${^{\vee}}X$ (called its right and left dual) and associated morphisms

$b_X:\mathbb{1}\to X\otimes X^{\vee}, d_X: X^{\vee}\otimes X\to \mathbb{1}$
$b_X:\mathbb{1}\to {^{\vee}}X\otimes X, d_X: X\otimes {^{\vee}}X\to \mathbb{1}$

obeying the zig-zag identities.

A twist on rigid braided monoidal category is a natural isomorphism from the identity functor to itself, with components $\theta_X \colon X \to X$ for which

$\theta_{X\otimes Y}=\beta_{Y,X}\beta_{X,Y}\theta_{X}\otimes \theta_{Y},$
$\theta_{\mathbb{1}}=\mathrm{id},$
$\theta_{X^{\vee}}=\theta_{X}^{\vee}.$

A ribbon category is a rigid braided monoidal category equipped with such a twist.

## Reference

Last revised on May 24, 2023 at 15:44:14. See the history of this page for a list of all contributions to it.