In mathematics, a ''commutativity constraint''

\gamma

on a

monoidal category In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left ...

\mathcal

'' is a choice of

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

\gamma_ : 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

. A braided monoidal category is a monoidal category

\mathcal

equipped with a braiding—that is, a commutativity constraint

\gamma

that satisfies axioms including the hexagon identities defined below. The term ''braided'' references the fact that the

braid group A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strande ...

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 In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some ...

. Alternatively, a braided monoidal category can be seen as a

tricategory In mathematics, a tricategory is a kind of structure of category theory studied in higher-dimensional category theory. Whereas a weak 2-category is said to be a '' bicategory'', a weak 3-category is said to be a ''tricategory'' (Gordon, Power ...

with one 0-cell and one 1-cell. Braided monoidal categories were introduced by

André Joyal André Joyal (; born 1943) is a professor of mathematics at the Université du Québec à Montréal who works on category theory. He was a member of the School of Mathematics at the Institute for Advanced Study in 2013, where he was invited to j ...

and

Ross Street Ross Howard Street (born 29 September 1945, Sydney) is an Australian mathematician specialising in category theory.monoidal structure on

\mathcal

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

, satisfy various

coherence condition In mathematics, and particularly category theory, a coherence condition is a collection of conditions requiring that various compositions of elementary morphisms are equal. Typically the elementary morphisms are part of the data of the category ...

s, 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

. In particular,

(\gamma_ \otimes \text) \circ (\text \otimes \gamma_) \circ (\gamma_ \otimes \text) =
(\text \otimes \gamma_) \circ (\gamma_ \otimes \text) \circ (\text \otimes \gamma_)

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_ \circ \gamma_ = \text

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 In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...

Ribbon categories

A braided monoidal category is a ''

ribbon category In mathematics, a ribbon category, also called a tortile category, is a particular type of braided monoidal category. Definition A monoidal category \mathcal C is, loosely speaking, a category equipped with a notion resembling the tensor produc ...

'' if it is rigid, and it may preserve quantum trace and co-quantum trace. Ribbon categories are particularly useful in constructing

Coboundary monoidal categories

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

(C, \otimes, \text)

together with a family of natural isomorphisms

\gamma_: A\otimes B \to B\otimes A

with the following properties: *

\gamma_ \circ \gamma_ = \text

for all pairs of objects

A

and

B

. *

\gamma_ \circ (\gamma_ \otimes \text)  = \gamma_ \circ (\text \otimes \gamma_)

The first property shows us that

\gamma^_ = \gamma_

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

Examples

* The category of representations of a group (or a

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...

) 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)

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 In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some ...

. * Symmetric

closed monoidal categories In mathematics, especially in category theory, a closed monoidal category (or a ''monoidal closed category'') is a category that is both a monoidal category and a closed category in such a way that the structures are compatible. A classic exampl ...

are used in denotational models of

linear logic Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. Although the logic has also ...

and

linear types Substructural type systems are a family of type systems analogous to substructural logics where one or more of the structural rules are absent or only allowed under controlled circumstances. Such systems are useful for constraining access to sy ...

. *Description and classification of topological ordered quantum systems.

References

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

* {{nlab, id=braided+monoidal+category, title=Braided monoidal category *

John Baez John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, appl ...

(1999)
An introduction to braided monoidal categories
''This week's finds in mathematical physics'' 137. Braids Monoidal categories ru:Симметричная моноидальная категория#Моноидальные категории с заузливанием