nLab
symmetric monoidal category

Idea

A symmetric monoidal category is a category with a product operation – a monoidal category – for which the product is as commutative as possible.

The point is that there are different degrees to which higher categorical products may be commutative. While a bare monoid is either commutative or not, a monoidal category may be a braided monoidal category – which already means that the order of products may be reversed up to some isomorphism – without being symmetric monoidal – which means that changing the order of a product twice, from $a\otimes b$ to $b\otimes a$ back to $a\otimes b$, indeed does yield a result equal to the original.

For higher monoidal categories there are accordingly ever more shades of the notion of “commutativity” of the monoidal product. This is described in detail at k-tuply monoidal n-category.

In general, the term symmetric monoidal is used for the maximally commutative case. See for instance symmetric monoidal (∞,1)-category. Notably, a symmetric monoidal ∞-groupoid is, under the homotopy hypothesis, the same as a connective spectrum.

Definition

A symmetric monoidal category is a braided monoidal category for which the braiding

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

satisfies an additional axiom:

$$B_{y,x}{B}_{x,y}=1_{x\otimes y}$$B_{y,x} B_{x,y} = 1_{x \otimes y}

for all objects $x,y$. Intuitively this says that switching things twice has no effect.

Expanding this out a bit: a symmetric monoidal category is, to begin with a category $M$ equipped with a functor

$$\otimes :M\times M\to M$$\otimes : M \times M \to M

called the tensor product, an object

$$1\in M$$1 \in M

called the unit object, a natural isomorphism

$$a_{x,y,z}:(x\otimes y)\otimes z\to x\otimes (y\otimes z)$$a_{x,y,z} : (x \otimes y) \otimes z \to x \otimes (y \otimes z)

called the associator, a natural isomorphism

$${\lambda }_x:1\otimes x\to x$$\lambda_x : 1 \otimes x \to x

called the left unitor, a natural isomorphism

$${\rho }_x:x\otimes 1\to x$$\rho_x : x \otimes 1 \to x

called the right unitor, and 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. We then demand that the associator obey the pentagon identity, which says this diagram commutes:

We demand that the associator and unitors obey the triangle identity, which says this diagram commutes:

$$\begin{array}{cccc}& (x\otimes 1)\otimes y& \stackrel{a_{x\mathrm{,1},y}}{⟶}& x\otimes (1\otimes y)\\ & {}_{{\rho }_x\otimes 1_y}\searrow & & {\swarrow }_{1_x\otimes {\lambda }_y}& \\ & & x\otimes y& & \end{array}$$\array{ & (x \otimes 1) \otimes y &\stackrel{a_{x,1,y}}{\longrightarrow} & x \otimes (1 \otimes y) \\ & {}_{\rho_x \otimes 1_y}\searrow && \swarrow_{1_x \otimes \lambda_y} & \\ && x \otimes y && }

We demand that the braiding and associator obey the first hexagon identity:

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

And lastly, we demand that

$$B_{y,x}{B}_{x,y}=1_{x\otimes y}.$$B_{y,x} B_{x,y} = 1_{x \otimes y} .

(The definition of braided monoidal category has two hexagon identities, but either one implies the other given this equation.)

Basic facts

There is a strict 2-category SymmMonCat with:

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

A symmetric monoidal category is equivalently a category that is equipped with the structure of an algebra over the little k-cubes operad for $k\ge 3$

Details are in examples 1.2.3 and 1.2.4 of

• Jacob Lurie, $𝔼[k]$-Algebras

Examples

• Every cartesian monoidal category is necessarily symmetric monoidal, due to the essential uniqueness of the categorical product. This includes cases such as Set, Cat.

• For $k$ some field, the category Vect of $k$-vector spaces carries the standard structure of a monoidal category coming from the tensor product, over $k$, of vector spaces. The standard braiding that identifies $V\otimes W$ with $W\otimes V$ by mapping homogeneous elements $v\otimes w$ to $w\otimes v$ obviously makes Vect into a symmetric monoidal category.

• The category of ${\mathbb{Z}}_2$-graded vector spaces, on the other hand, has two different symmetric monoidal extensions of the standard tensor product monoidal structure. One is the trivial one from above, the other is the one that induces a a sign when two odd-graded vectors $v$ and $w$ are passed past each other : $v\otimes w\mapsto -w\otimes v$. This non-trivial symmetric monoidal structure on Vect[\mathbb[Z}_2] defines the symmetric monoidal category of super vector spaces.

References

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

• John Baez, Some definitions everyone should know.

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

• John Baez and Mike Stay, Physics, topology, logic and computation: a Rosetta Stone.

Eventually we should include all these definitions and diagrams here! I don’t know a good way to do this yet.