nLab ribbon category

Contents

Context

Monoidal categories

monoidal categories

With symmetry

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 (also called a tortile category or balanced rigid braided tensor category) is a monoidal category (π’ž,βŠ—,πŸ™,Ξ±,l,r)(\mathcal{C}, \otimes, \mathbb{1}, \alpha, l, r) equipped with braiding Ξ²={Ξ² X,Y}\beta=\{\beta_{X,Y}\}, twist ΞΈ={ΞΈ X}\theta=\{\theta_X\} and duality (∨,b,d)(\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 π’ž\mathcal{C} equipped with a braiding Ξ²\beta, which is a natural isomorphisms Ξ² X,Y:XβŠ—Yβ†’YβŠ—X\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 XX, there exist objects X ∨X^{\vee} and ∨X{^{\vee}}X (called its right dual and left dual) and associated morphisms

b X:πŸ™β†’XβŠ—X ∨,d X:X βˆ¨βŠ—Xβ†’πŸ™b_X:\mathbb{1}\to X\otimes X^{\vee}, d_X: X^{\vee}\otimes X\to \mathbb{1}
b X:πŸ™β†’βˆ¨XβŠ—X,d X:XβŠ—βˆ¨Xβ†’πŸ™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 θ X:X→X\theta_X \colon X \to X for which

ΞΈ XβŠ—Y=Ξ² Y,XΞ² X,YΞΈ XβŠ—ΞΈ Y,\theta_{X\otimes Y}=\beta_{Y,X}\beta_{X,Y}\theta_{X}\otimes \theta_{Y},
ΞΈ πŸ™=id,\theta_{\mathbb{1}}=\mathrm{id},
θ X ∨=θ X ∨.\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

Last revised on June 7, 2022 at 08:03:42. See the history of this page for a list of all contributions to it.