Braided monoidal category

In mathematics, a commutativity constraint on a monoidal category ${\mathcal {C}}$ is a natural isomorphism $\gamma$ from ${\mathcal {C}}$ to ${\mathcal {C}}^{op}$, where ${\mathcal {C}}^{op}$ is the category with the opposite tensor product. Explicitly, $\gamma$ 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 commutativity constraint $\gamma$ which satisfies the hexagon axiom (see below). The term braided comes from 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 various related notions are important 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 axiom

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:

• The braiding commutes with the units. That is, the following diagram commutes:

• The action of $\gamma$ on an $N$-fold tensor product factors through the braid group. 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

See also: Strongly ribbon category

A braided monoidal category is a ribbon category if it is rigid, and it has a good notion of 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:

• $\gamma _{B,A}\circ \gamma _{A,B}={\text{Id}}$ for all pairs of objects $A$ and $B$.

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

Examples

• The category of representations of a group (or a lie algebra) is a symmetric monoidal category where $\gamma (v\otimes w)=w\otimes v$.

• The category of representations of a quantized universal enveloping algebra $U_{q}({\mathfrak {g}})$ is a braided monoidal category, where $\gamma$ is constructed using the Universal R-matrix. In fact, this example is a ribbon category as well.

Applications

• knot invariants.
• Symmetric closed monoidal categories are used in denotational models of linear logic and linear types.

References

• Chari, Vyjayanthi; Pressley, Andrew. "A guide to quantum groups". Cambridge University Press. 1995.

• Joyal, André; Street, Ross (1993). "Braided Tensor Categories". Advances in Mathematics 102, 20–78.

• Savage, Alistair. Braided and coboundary monoidal categories. Algebras, representations and applications, 229--251, Contemp. Math., 483, Amer. Math. Soc., Providence, RI, 2009. Available on the arXiv

External links

• John Baez (1999), An introduction to braided monoidal categories, This week's finds in mathematical physics 137.