nLab braided monoidal category

Idea

Intuitively speaking, a braided monoidal category is a category with a tensor product and an isomorphism called the ‘braiding’ which lets us ‘switch’ two objects in a tensor product like $x \otimes y$. Thus the tensor product is “commutative” in a sense, but not as coherently commutative as in a symmetric monoidal category.

A braided monoidal category is a special case of the notion of braided pseudomonoid in a braided monoidal 2-category.

Definition

Definition

A braided monoidal category, or (“braided tensor category”, but see there), is a monoidal category $\mathcal{C}$ equipped with a natural isomorphism

$$B_{x,y} : x \otimes y \to y \otimes x$$

called the braiding, such that the following two kinds of diagrams commute for all objects involved (called the hexagon identities encoding the compatibility of the braiding with the associator for the tensor product):

$$\array{ (x \otimes y) \otimes z &\stackrel{a_{x,y,z}}{\to}& x \otimes (y \otimes z) &\stackrel{B_{x,y \otimes z}}{\to}& (y \otimes z) \otimes x \\ \downarrow^{B_{x,y}\otimes Id} &&&& \downarrow^{a_{y,z,x}} \\ (y \otimes x) \otimes z &\stackrel{a_{y,x,z}}{\to}& y \otimes (x \otimes z) &\stackrel{Id \otimes B_{x,z}}{\to}& y \otimes (z \otimes x) }$$

and

$$\array{ x \otimes (y \otimes z) &\stackrel{a^{-1}_{x,y,z}}{\to}& (x \otimes y) \otimes z &\stackrel{B_{x \otimes y, z}}{\to}& z \otimes (x \otimes y) \\ \downarrow^{Id \otimes B_{y,z}} &&&& \downarrow^{a^{-1}_{z,x,y}} \\ x \otimes (z \otimes y) &\stackrel{a^{-1}_{x,z,y}}{\to}& (x \otimes z) \otimes y &\stackrel{B_{x,z} \otimes Id}{\to}& (z \otimes x) \otimes y } \,,$$

where $a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z)$ denotes the components of the associator of $\mathcal{C}^\otimes$.

Definition

If the braiding in def. “squares” to the identity in that $B_{y,x} \circ B_{x,y} = id_{x \otimes y}$, then the braided monoidal category is called a symmetric monoidal category.

Remark

Intuitively speaking, the first hexagon identity in def. says we may braid $x$ past $y \otimes z$ ‘all at once’ or in two steps. The second hexagon identity says that we may braid $x \otimes y$ past $z$ all at once or in two steps.

Remark

From these axioms in def. , it follows that the braiding is compatible with the left and right unitors $l_x : I \otimes x \to x$ and $r_x : x \otimes I \to x$. That is to say, for all objects $x$ the diagram

$$\array{ I \otimes x &&\stackrel{B_{I,x}}{\to}&& x \otimes I \\ & {}_{l_x}\searrow && \swarrow_{r_x} \\ && x }$$

commutes.

In terms of higher monoidal structure

In terms of the language of k-tuply monoidal n-categories a braided monoidal category is a doubly monoidal 1-category .

Accordingly, by delooping twice, it may be identified with a tricategory with a single object and a single 1-morphism.

However, unlike the definition of a monoidal category as a bicategory with one object, this identification is not trivial; a doubly-degenerate tricategory is literally a category with two monoidal structures that interchange up to isomorphism. It requires the Eckmann-Hilton argument to deduce an equivalence with braided monoidal categories.

A commutative monoid is the same as a monoid in the category of monoids. Similarly, a braided monoidal category is equivalent to a monoidal-category object (that is, a pseudomonoid) in the monoidal 2-category of monoidal categories. This result goes back to the 1986 paper by Joyal and Street. (There is also a notion of braided pseudomonoid that specializes directly in Cat to braided monoidal categories.)

A braided monoidal category is equivalently a category that is equipped with the structure of an algebra over the little 2-cubes operad.

Details are in example 1.2.4 of

• Jacob Lurie, $\mathbb{E}[k]$-Algebras

The 2-category of braided monoidal categories

There is a strict 2-category BrMonCat with:

• braided monoidal categories as objects,
• braided monoidal functors as morphisms,
• braided monoidal natural transformations as 2-morphisms.

Examples

• Any symmetric monoidal category is a braided monoidal category.

• The monoidal category $R$-$\mathcal{GMod}$ of graded modules over a commutative ring $R$ (with the usual tensor product of graded modules) can be made into a braided monoidal category in multiple ways, each one given by an invertible element $u$ of the base ring. Infact, all braidings on $R$-$\mathcal{GMod}$ arise in this way (as in Joyal, Street). The braiding $B^u_{V,W} : V \otimes W \to W \otimes V$, is defined as

  $$x \otimes y \mapsto u^{|x||y|} y \otimes x,$$

  where $|x|$ and $|y|$ denote the degrees. It’s evident that the resulting braided monoidal category is symmetric if and only if $u^2 = 1$.

• The category of crossed G-sets.

• The braid category is the free braided strict monoidal category on an object.

• The category of vines is the free braided strict monoidal category containing a braided monoid.

Properties

Tannaka duality

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebra	category of modules
$A$	$Mod_A$
$R$-algebra	$Mod_R$-2-module
sesquialgebra	2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebra	strict 2-ring: monoidal category with fiber functor
Hopf algebra	rigid monoidal category with fiber functor
hopfish algebra (correct version)	rigid monoidal category (without fiber functor)
weak Hopf algebra	fusion category with generalized fiber functor
quasitriangular bialgebra	braided monoidal category with fiber functor
triangular bialgebra	symmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)	rigid braided monoidal category with fiber functor
triangular Hopf algebra	rigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)	rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld double	form Drinfeld center
trialgebra	Hopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category	2-category of module categories
$A$	$Mod_A$
$R$-2-algebra	$Mod_R$-3-module
Hopf monoidal category	monoidal 2-category (with some duality and strictness structure)

3-Tannaka duality for module 2-categories over monoidal 2-categories

monoidal 2-category	3-category of module 2-categories
$A$	$Mod_A$
$R$-3-algebra	$Mod_R$-4-module

Related concepts

• k-tuply monoidal (n,r)-category

• monoidal category, monoidal (∞,1)-category

• braided monoidal category, braided monoidal (∞,1)-category

  • modular tensor category

  • braided monoidal dagger category

  • braided monoidal 2-category

• symmetric monoidal category, symmetric monoidal (∞,1)-category

• closed monoidal category , closed monoidal (∞,1)-category

• 2-fold monoidal category, duoidal category

• braided monoidal groupoid

  • braided 2-group, braided ∞-group

References

Textbook accounts:

• Saunders MacLane, §XI of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

• Francis Borceux, Section 6.1 of: Handbook of Categorical Algebra Vol. 2: Categories and Structures $[$doi:10.1017/CBO9780511525865$]$, Encyclopedia of Mathematics and its Applications 50, Cambridge University Press (1994)

• Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, Chapter 2 of: Tensor Categories, AMS Mathematical Surveys and Monographs 205 (2015) $[$ISBN:978-1-4704-3441-0, pdf$]$

  (focus on tensor categories)

Exposition of basics of monoidal categories and categorical algebra:

• geometry of physics – categories and toposes, Section 2: Basic notions of categorical algebra

The original articles on braided monoidal categories (the published version does not completely supersede the Macquarie Math Reports, which have some extra results):

• André Joyal, Ross Street: Braided monoidal categories, Macquarie Math Reports 85-0067 (1985) [pdf, pdf]

• André Joyal, Ross Street: Braided monoidal categories, Macquarie Math Reports 86-0081 (1986) [pdf, pdf]

• André Joyal, Ross Street, Braided tensor categories, Adv. Math. 102 (1993) 20-78 [doi:10.1006/aima.1993.1055]

Around the same time the same definition was also proposed independently by Lawrence Breen in a letter to Pierre Deligne:

• Lawrence Breen, Une lettre à P. Deligne au sujet des $2$-catégories tressées (1988) (pdf)

Exposition of basic definitions:

• John Baez, Some definitions everyone should know (2004)

An elementary introduction using string diagrams:

• John Baez, Mike Stay: Physics, Topology, Logic and Computation: A Rosetta Stone, in: New Structures for Physics, Lecture Notes in Physics 813 Springer (2011) 95-174 [arXiv:0903.0340, doi:10.1007/978-3-642-12821-9_2]

A generalisation of braidings to lax braidings, where $B$ is not required to be invertible:

• Brian Day, Elango Panchadcharam, Ross Street, Lax braidings and the lax centre, Contemporary Mathematics 441 1 (2007) [pdf, pdf]

On a construction of group-crossed tensor categories that depends on both a group action and a grading:

• Mizuki Oikawa. Center construction for group-crossed tensor categories (2024). (arXiv:2404.09972).

On a reformulation of braided monoidal categories in a manner closer to symmetric closed categories and braided skew-monoidal categories:

• Alexei Davydov, Ingo Runkel: An alternative description of braided monoidal categories, Applied Categorical Structures 23 (2015) 279-309 [doi:10.1007/s10485-013-9338-3, pdf]