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 $x\otimes y$.

Definition

A braided monoidal category, or braided tensor category, is a monoidal category $V$ equipped with a natural isomorphism

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

called the braiding, which must satisfy two axioms called hexagon identities encoding the compatibility of the braiding with the associator for the tensor product.

If the braiding squares to the identity, then the braided monoidal category is a symmetric monoidal category.

The coherence laws

To see the haxagon identities, let us write $a_{x,y,z}:(x\otimes y)\otimes z\to x\otimes (y\otimes z)$ for the components of the associator in $V$. Then the first hexagon identity says that for all $x,y,z\in \mathrm{Obj}(V)$ the following diagram commutes:

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

The second hexagon identity says that for all $x,y,z\in \mathrm{Obj}(V)$ the following diagram commutes:

$$\begin{array}{ccccc}x\otimes (y\otimes z)& \stackrel{a_{x,y,z}^{-1}}{\to }& (x\otimes y)\otimes z& \stackrel{B_{x\otimes y,z}}{\to }& z\otimes (x\otimes y)\\ {\downarrow }^{\mathrm{Id}\otimes B_{y,z}}& & & & {\downarrow }^{a_{z,x,y}^{-1}}\\ x\otimes (z\otimes y)& \stackrel{a_{x,z,y}^{-1}}{\to }& (x\otimes z)\otimes y& \stackrel{B_{x,z}\otimes \mathrm{Id}}{\to }& (z\otimes x)\otimes y\end{array}$$\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 }

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

From these axioms, 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

$$\begin{array}{ccccc}I\otimes X& & \stackrel{B_{I,x}}{\to }& & x\otimes I\\ & {}_{l_x}\searrow & & {\swarrow }_{r_x}\\ & & X\end{array}$$\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 probably goes back to Joyal and Street.

Reference, anyone?

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, $𝔼[k]$-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.

References

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:

• André Joyal and Ross Street, Braided monoidal categories, Macquarie Math Reports 860081 (1986).

• André Joyal and Ross Street, Braided tensor categories , Adv. Math. 102 (1993), 20–78.

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

• John Baez, Some definitions everyone should know.

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

• John Baez and Mike Stay, Physics, topology, logic and computation: a Rosetta Stone, to appear in New Structures in Physics, ed. Bob Coecke.

Eventually we should include all these diagrams here, along with the definition of braided monoidal functor and braided monoidal natural transformation! Can anyone help out?