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:

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

Properties

The 2-category of symmetric monoidal categories

There is a strict 2-category $\mathrm{SymmMonCat}$ with:

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

As models for connective spectra

The nerve of a symmetric monoidal category is always an infinite loop space, hence the degree-0-space of a connective spectrum. One calls this also the K-theory spectrum of the symmetric monoidal category:

$$\begin{array}{ccccc}& & & & \mathrm{Spectra}\\ & & {}^K\nearrow & & \downarrow \\ \mathrm{SymmMonCat}& \to & \mathrm{Cat}& \underset{N}{\to }& \mathrm{sSet}\end{array}.$$\array{ &&&& Spectra \\ && {}^{\mathllap{K}}\nearrow && \downarrow \\ SymmMonCat &\to& Cat &\underset{N}{\to}& sSet } \,.

This construction extended to an equivalence of categories

$$K:\mathrm{Ho}(\mathrm{SymmMonCat})\stackrel{\simeq }{\to }\mathrm{Ho}(\mathrm{Spectra}{)}_{\ge 0}\hookrightarrow \mathrm{Ho}(\mathrm{Spectra})$$K : Ho(SymmMonCat) \stackrel{\simeq}{\to} Ho(Spectra)_{\geq 0} \hookrightarrow Ho(Spectra)

between the full subcategory of the stable homotopy category $\mathrm{Ho}(\mathrm{Spectra})$ on the connective spectra and the homotopy category of $\mathrm{SymmMonCat}$, regarded with the transferred structure of a category with weak equivalences.

This is due to (Thomason, 95). Further discussion is in (Mandell, 2010).

Notice that this is almost the complete analog in stable homotopy theory of the Quillen equivalence between the Thomason model structure on Cat and the standard model structure on simplicial sets. Only that $\mathrm{SymmMonCat}$ cannot carry a model category structure because it does not have all colimits. In some sense the “colimit completion” of $\mathrm{SymmMonCat}$ is the category of multicategories. Once expects that this carries a model structure that refines the above equivalence of homotopy categories to a Quillen equivalence.

(This is currently being investigated by Elmendorf, Nikolaus and maybe others.)

As algebras over the little $k$-cubes operad

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.

Tannaka duality

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebra	category of modules
$A$	${\mathrm{Mod}}_A$
$R$-algebra	${\mathrm{Mod}}_R$-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
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
$A$	${\mathrm{Mod}}_A$
$R$-2-algebra	${\mathrm{Mod}}_R$-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
$A$	${\mathrm{Mod}}_A$
$R$-3-algebra	${\mathrm{Mod}}_R$-4-module

Related concepts

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

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

  • permutative category, bipermutative category

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

References

For a survey of definitions of symmetric monoidal categories, symmetric monoidal functors and symmetric monoidal natural transformations, see for instance

• 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.

The theorem that symmetric monoidal categories model all connective spectra is due to

• R. Thomason, Symmetric monoidal categories model all connective spectra , Theory and applications of Categories, Vol. 1, No. 5, (1995) pp. 78-118

More discussion is in

• Michael Mandell, An Inverse K-Theory Functor (arXiv:1002.3622)