mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a ''commutativity constraint''

\gamma

on a

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

\mathcal

'' is a choice of

isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism 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 the ...

\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 In mathematics, the braid group on strands (denoted B_n), also known as the Artin braid group, is the group whose elements are equivalence classes of Braid theory, -braids (e.g. under ambient isotopy), and whose group operation is composition of ...

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, especially in category theory, a 3-category is a 2-category together with 3-morphisms. It comes in at least three flavors *a strict 3-category, *a semi-strict 3-category also called a Gray category, *a weak 3-category. The coherence ...

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 jo ...

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, specifically in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or "up to isomorphism". The adjectives such as "pseudo-" and "lax-" ...

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 grou ...

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 product ...

'' 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 ident ...

) 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 example ...

are used in denotational models of

linear logic Linear logic is a substructural logic proposed by French logician 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 ...

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 can constrain access to system resources ...

. *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, ap ...

(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