Braided monoidal category

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In mathematics, a commutativity constraint  \gamma on a monoidal category \mathcal{C} 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[edit]

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}:

CategoryBraiding-02.png CategoryBraiding-03.png

Properties[edit]

Coherence[edit]

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:
CategoryBraiding-01.png
  • 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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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

Applications[edit]

References[edit]

  • 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[edit]