Braided monoidal category

In mathematics, a "commutativity constraint" formula_1 on a monoidal category "formula_2" is a choice of isomorphism formula_3 for each pair of objects "A" and "B" which form a "natural family." In particular, to have a commutativity constraint, one must have formula_4 for all pairs of objects formula_5.
A braided monoidal category is a monoidal category formula_2 equipped with a braiding—that is, a commutativity constraint formula_1 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.
Braided monoidal categories were introduced by André Joyal and Ross Street in a 1986 preprint. A modified version of this paper was published in 1993.
The hexagon identities.
For formula_2 along with the commutativity constraint formula_9 to be called a braided monoidal category, the following hexagonal diagrams must commute for all objects formula_10. Here formula_11 is the associativity isomorphism coming from the monoidal structure on formula_2:
Properties.
Coherence.
It can be shown that the natural isomorphism formula_9 along with the maps formula_14 coming from the monoidal structure on the category formula_2, satisfy various coherence conditions, which state that various compositions of structure maps are equal. In particular:
formula_18
as maps formula_19. 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 formula_9 also satisfies formula_21 for all pairs of objects formula_22 and formula_23. In this case the action of formula_9 on an formula_17-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 formula_26 together with a family of natural isomorphisms formula_27 with the following properties:
The first property shows us that formula_32, thus allowing us to omit the analog to the second defining diagram of a braided monoidal category and ignore the associator maps as implied.