nLab ribbon category

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.

Related concepts

• modular tensor category

Reference

• Nicolai Reshetikhin, Vladimir Turaev, Ribbon graphs and their invariants derived from quantum groups, Commun. Math. Phys. 127 1 (1990) [doi:10.1007/BF02096491]

• André Joyal, Ross Street, Braided tensor categories, Advances in Mathematics 102 (1993) 20–78 [doi:10.1006/aima.1993.1055]

• Mei Chee Shum, Tortile tensor categories, Journal of Pure and Applied Algebra 93 1 (1994) 57-110 [10.1016/0022-4049(92)00039-T]