Braided monoidal category
In mathematics, a commutativity constraint on a monoidal category is a choice of isomorphism for each pair of objects A and B which form a "natural family." In particular, to have a commutativity constraint, one must have for all pairs of objects .
A braided monoidal category is a monoidal category equipped with a braiding—that is, a commutativity constraint that satisfies axioms including the hexagon identities 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 along with the commutativity constraint to be called a braided monoidal category, the following hexagonal diagrams must commute for all objects . Here is the associativity isomorphism coming from the monoidal structure on :
It can be shown that the natural isomorphism along with the maps coming from the monoidal structure on the category , 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 on an -fold tensor product factors through the braid group. In particular,
as maps . Here we have left out the associator maps.
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 also satisfies for all pairs of objects and . In this case the action of on an -fold tensor product factors through the symmetric group.
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:
- for all pairs of objects and .
- The category of representations of a group (or a Lie algebra) is a symmetric monoidal category where .
- The category of representations of a quantized universal enveloping algebra is a braided monoidal category, where is constructed using the universal R-matrix. In fact, this example is a ribbon category as well.