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."

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.

Braided monoidal categories were introduced by André Joyal and Ross Street in a 1986 preprint.

[1] A modified version of this paper was published in 1993.

to be called a braided monoidal category, the following hexagonal diagrams must commute for all objects

is the associativity isomorphism coming from the monoidal structure on

: It can be shown that the natural isomorphism

α , λ , ρ

coming from the monoidal structure on the category

, satisfy various coherence conditions, which state that various compositions of structure maps are equal.

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.

A braided monoidal category is called symmetric if

for all pairs of objects

-fold tensor product factors through the symmetric group.

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.

A coboundary or “cactus” monoidal category is a monoidal category

together with a family of natural isomorphisms

, thus allowing us to omit the analog to the second defining diagram of a braided monoidal category and ignore the associator maps as implied.