nLab braided monoidal category



Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



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 xyx \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.



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

B x,y:xyyx 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):

(xy)z a x,y,z x(yz) B x,yz (yz)x B x,yId a y,z,x (yx)z a y,x,z y(xz) IdB x,z y(zx) \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) }


x(yz) a x,y,z 1 (xy)z B xy,z z(xy) IdB y,z a z,x,y 1 x(zy) a x,z,y 1 (xz)y B x,zId (zx)y, \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:(xy)zx(yz)a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z) denotes the components of the associator of 𝒞 \mathcal{C}^\otimes.


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


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


From these axioms in def. , it follows that the braiding is compatible with the left and right unitors l x:Ixxl_x : I \otimes x \to x and r x:xIxr_x : x \otimes I \to x. That is to say, for all objects xx the diagram

Ix B I,x xI l x r x x \array{ I \otimes x &&\stackrel{B_{I,x}}{\to}&& x \otimes I \\ & {}_{l_x}\searrow && \swarrow_{r_x} \\ && x }


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

The 2-category of braided monoidal categories

There is a strict 2-category BrMonCat with:



Tannaka duality

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebracategory of modules
AAMod AMod_A
RR-algebraMod RMod_R-2-module
sesquialgebra2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebrastrict 2-ring: monoidal category with fiber functor
Hopf algebrarigid monoidal category with fiber functor
hopfish algebra (correct version)rigid monoidal category (without fiber functor)
weak Hopf algebrafusion category with generalized fiber functor
quasitriangular bialgebrabraided monoidal category with fiber functor
triangular bialgebrasymmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)rigid braided monoidal category with fiber functor
triangular Hopf algebrarigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld doubleform Drinfeld center
trialgebraHopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category2-category of module categories
AAMod AMod_A
RR-2-algebraMod RMod_R-3-module
Hopf monoidal categorymonoidal 2-category (with some duality and strictness structure)

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

monoidal 2-category3-category of module 2-categories
AAMod AMod_A
RR-3-algebraMod RMod_R-4-module


Textbook accounts:

Exposition of basics of monoidal categories and categorical algebra:

The original papers on braided monoidal categories are by Joyal and Street. The published version does not completely supersede the Macquarie Math Reports version, which has some nice extra results:

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 22-catégories tressées (1988) (pdf)

Textbook accounts include

For a review of definitions of braided monoidal categories, braided monoidal functors and braided monoidal natural transformations, see:

For an elementary introduction to braided monoidal categories using string diagrams, see:

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

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).

Last revised on April 16, 2024 at 08:58:41. See the history of this page for a list of all contributions to it.