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 $(\vee, b, d)$ that satisfy some compatibility conditions. The name ribbon category was introduced by Reshetikhin and Turaev in their work in 1990, the name tortile category was used by Joyal and Street in their work.

Definition

A braided monoidal category is a monoidal category $\mathcal{C}$ equipped with a braiding $\beta$, which is a natural isomorphisms $\beta_{X,Y}\colon X \otimes Y \to Y \otimes X$ 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 dual 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 set of isomorphisms $\theta_X \colon X \to X$ 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 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