Braided monoidal category

The hexagon identities

For \mathcal{C} along with the commutativity constraint \gamma to be called a braided monoidal category, the following hexagonal diagrams must commute for all objects A,B,C \in \mathcal{C}. Here \alpha is the associativity isomorphism coming from the monoidal structure on \mathcal{C}:

Properties

Coherence

It can be shown that the natural isomorphism \gamma along with the maps \alpha, \lambda, \rho coming from the monoidal structure on the category \mathcal{C}, satisfy various coherence conditions, which state that various compositions of structure maps are equal. In particular:

• The braiding commutes with the units. That is, the following diagram commutes:

• The action of \gamma on an N-fold tensor product factors through the braid group. In particular,

(\gamma_{B,C} \otimes \text{Id}) \circ (\text{Id} \otimes \gamma_{A, C}) \circ (\gamma_{A,B} \otimes \text{Id}) = (\text{Id} \otimes \gamma_{A,B}) \circ (\gamma_{A,C} \otimes \text{Id}) \circ (\text{Id} \otimes \gamma_{B, C})

as maps A \otimes B \otimes C \rightarrow C \otimes B \otimes A. Here we have left out the associator maps.

Variations

There are several variants of braided monoidal categories that are used in various contexts. See, for example, the expository paper of Savage (2009) for an explanation of symmetric and coboundary monoidal categories, and the book by Chari and Pressley (1995) for ribbon categories.

Symmetric monoidal categories

Main article: Symmetric monoidal category

A braided monoidal category is called symmetric if \gamma also satisfies \gamma_{B,A} \circ \gamma_{A,B} = \text{Id} for all pairs of objects A and B. In this case the action of \gamma on an N-fold tensor product factors through the symmetric group.

Ribbon categories

A braided monoidal category is a ribbon category if it is rigid, and it may preserve quantum trace and co-quantum trace. Ribbon categories are particularly useful in constructing knot invariants.

Coboundary monoidal categories

A coboundary or “cactus” monoidal category is a monoidal category (C, \otimes, \text{Id}) together with a family of natural isomorphisms \gamma_{A,B}: A\otimes B \to B\otimes A with the following properties:

• \gamma_{B,A} \circ \gamma_{A,B} = \text{Id} for all pairs of objects A and B.
• \gamma_{B \otimes A, C} \circ (\gamma_{A,B} \otimes \text{Id}) = \gamma_{A, C \otimes B} \circ (\text{Id} \otimes \gamma_{B,C})

The first property shows us that \gamma^{-1}{A,B} = \gamma{B,A} , thus allowing us to omit the analog to the second defining diagram of a braided monoidal category and ignore the associator maps as implied.

Examples

• The category of representations of a group (or a Lie algebra) is a symmetric monoidal category where \gamma (v \otimes w) = w \otimes v .
• The category of representations of a quantized universal enveloping algebra U_q(\mathfrak{g}) is a braided monoidal category, where \gamma is constructed using the universal R-matrix. In fact, this example is a ribbon category as well.

Applications

• Knot invariants.
• Symmetric closed monoidal categories are used in denotational models of linear logic and linear types.
• Description and classification of topological ordered quantum systems.

References

• Chari, Vyjayanthi; Pressley, Andrew. "A guide to quantum groups". Cambridge University Press. 1995.
• Savage, Alistair. Braided and coboundary monoidal categories. Algebras, representations and applications, 229–251, Contemp. Math., 483, Amer. Math. Soc., Providence, RI, 2009. Available on the arXiv

References

1. (November 1986). "Braided monoidal categories". Macquarie Mathematics Reports.
2. (1993). "Braided tensor categories". [[Advances in Mathematics]].