nLab
ribbon category

Idea

A ribbon category (also called a tortile category 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 $(\vee, b, d)$ which satisfy some compatible conditions. The name ribbon category is introduced by Reshetikhin and Turaev in their work in 1990, the name tortile category is used by Joyal and Street in their work.

Definition

A braided monoidal tensor category is a monoidal category $\mathcal{C}$ equipped with a braiding $\beta=\{\beta_{X,Y}\}$ which is a set of isomorphisms natural in $X$ and $Y$ and satisfies the hexagon relation.

A braided tensor category $(\mathcal{C}, \otimes, \mathbb{1}, \alpha, l, r, \beta)$ is rigid if, for every object $X$, there exist objects $X^{\vee}$ and ${^{\vee}}X$ (called right dual and left dual) and associated morphsims

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

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

which satisfy some consistency conditions.

The twist on rigid braided tensor category is a set of isomorphism $\theta=\{\theta_X\}$ 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 Tensor category equipped with a twist.

Reference

• N. Y. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Commun.Math. Phys. (1990) 127: 1. https://doi.org/10.1007/BF02096491
• A. Joyal and R. Street, Braided tensor categories, Advances in Mathematics, 102: 20–78, doi:10.1006/aima.1993.1055