Braided monoidal category

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

A braided monoidal category is a monoidal category ${\displaystyle {\mathcal {C}}}$ equipped with a braiding—that is, a commutativity constraint ${\displaystyle \gamma }$ 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.

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

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

${\displaystyle (\gamma _{B,C}\otimes {\text{Id}})\circ ({\text{Id}}\otimes \gamma _{A,C})\circ (\gamma _{A,B}\otimes {\text{Id}})=({\text{Id}}\otimes \gamma _{A,B})\circ (\gamma _{A,C}\otimes {\text{Id}})\circ ({\text{Id}}\otimes \gamma _{B,C})}$

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