nLab ribbon category

1. Idea

A ribbon category [Reshetikhin & Turaev (1990)] (also called a tortile category [Joyal & Street (1993), Shum 1994, Selinger 2011 §4.7] 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.

2. Definition

Recall that:

Definition 2.1. 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.

Definition 2.2. A rigid monoidal category is a braided monoidal category (Def. 2.1) where for every object $X$, there exist objects $X^{\vee}$ and ${^{\vee}}X$ (called its right and left dual) and associated morphisms

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

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

obeying the zig-zag identities:

$$(d_X\otimes \text{id}_X)\circ (\text{id}_X\otimes b_{X})=\text{id}_{X},$$

$$(\text{id}_{X^{\vee}}\otimes d_{X})\circ ( b_{X}\otimes \text{id}_{X^{\vee}})=\text{id}_{X^{\vee}}.$$

Now:

Definition 2.3. A twist on rigid braided monoidal category (Def. 2.2) is a natural isomorphism from the identity functor to itself, with components $\theta_X \colon X \to X$ for which

$$\theta_{X\otimes Y} \;=\; \beta_{Y,X} \circ \beta_{X,Y} \circ \theta_{X}\otimes \theta_{Y} \,,$$

$$\theta_{\mathbb{1}} \;=\; \mathrm{id} \,,$$

$$\theta_{X^{\vee}} \;=\; \theta_{X}^{\vee} \,.$$

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

(e.g. Shum 1994 Def. 1.3).

A functor between ribbon categories is a ribbon functor (tortile functor) if it preserves all this structure up to isomorphism.

3. Properties

Relation to tangles

Proposition 3.1. (Shum's theorem)
The category of framed oriented tangles is equivalently the free ribbon category generated by a single object.

(Shum 1994, Yetter 2001 Thm. 9.1)

4. Related concepts

• modular tensor category

5. References

• 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]

• Vladimir Turaev, §I.1 in: Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics 18, de Gruyter & Co. (1994) [doi:10.1515/9783110435221, pdf]

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

• David N. Yetter: Functorial Knot Theory – Categories of Tangles, Coherence, Categorical Deformations, and Topological Invariants, Series on Knots and Everything 26, World Scientific (2001) [doi:10.1142/4542]

• Peter Selinger, §4.7 in: A survey of graphical languages for monoidal categories, Springer Lecture Notes in Physics 813 (2011) 289-355 [arXiv:0908.3347, doi:10.1007/978-3-642-12821-9_4]

Lecture notes:

• Daniel Bump: Ribbon categories, lecture 4 of Quantum Groups (2022) [slides: pdf, pdf, notes: pdf]