nLab ribbon category

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Contents

Context

Monoidal categories

monoidal categories

With braiding

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

Knot theory

Contents

Idea

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

Definition

Recall that:

Definition

A braided monoidal category is a monoidal category π’ž\mathcal{C} equipped with a braiding Ξ²\beta, which is a natural isomorphism Ξ² X,Y:XβŠ—Yβ†’YβŠ—X\beta_{X,Y}\colon X \otimes Y \to Y \otimes X obeying the hexagon identities.

Definition

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

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

obeying the zig-zag identities:

(d XβŠ—id X)∘(id XβŠ—b X)=id X,(d_X\otimes \text{id}_X)\circ (\text{id}_X\otimes b_{X})=\text{id}_{X},
(id X βˆ¨βŠ—d X)∘(b XβŠ—id X ∨)=id X ∨.(\text{id}_{X^{\vee}}\otimes d_{X})\circ ( b_{X}\otimes \text{id}_{X^{\vee}})=\text{id}_{X^{\vee}}.

Now:

Definition

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

ΞΈ XβŠ—Y=Ξ² Y,X∘β X,Y∘θ XβŠ—ΞΈ Y, \theta_{X\otimes Y} \;=\; \beta_{Y,X} \circ \beta_{X,Y} \circ \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 (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.

Properties

Relation to tangles

(Shum 1994, Yetter 2001 Thm. 9.1)

References

Lecture notes:

Last revised on August 31, 2024 at 19:41:12. See the history of this page for a list of all contributions to it.