Braided monoidal category

In mathematics, a commutativity constraint  \gamma on a monoidal category \mathcal{C} is a choice of isomorphism  \gamma_{A,B} : A\otimes B \rightarrow B\otimes A for each pair of objects A and B which form a "natural family." In particular, to have a commutativity constraint, one must have  A \otimes B \cong B \otimes A for all pairs of objects  A,B \in \mathcal{C}.

A braided monoidal category is a monoidal category \mathcal{C} equipped with a braiding - that is, a commutativity constraint  \gamma which satisfies the hexagon identities as defined below. The term braided references the fact that the braid group plays an important role in the theory of braided monoidal categories. Partly for this reason, braided monoidal categories and other topics are related in the theory of knot invariants.

Alternatively, a braided monoidal category can be seen as a tricategory with one 0-cell and one 1-cell.

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:

 (\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

A braided monoidal category is called symmetric if \gamma also satisfies  \gamma_{B,A} \circ \gamma_{A,B} = 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

Sometimes categories are assumed to have n-ary monoidal products for all finite n (in particular n>2), diminishing the role of associator morphisms. In such categories, the following variant is used, where the hexagon axiom is replaced by the two conditions:

Examples

Applications

References

External links

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