symmetric monoidal category


Monoidal categories



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 aba \otimes b to bab \otimes a back to aba \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.

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



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

B x,y:xyyx B_{x,y} \colon x \otimes y \to y \otimes x

satisfies the condition:

B y,xB x,y=1 xy B_{y,x} \circ B_{x,y} = 1_{x \otimes y}

for all objects x,yx, y

Intuitively this says that switching things twice in the same direction has no effect.

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

:M×MM \otimes : M \times M \to M

called the tensor product, an object

1M 1 \in M

called the unit object, a natural isomorphism

a x,y,z:(xy)zx(yz) a_{x,y,z} : (x \otimes y) \otimes z \to x \otimes (y \otimes z)

called the associator, a natural isomorphism

λ x:1xx \lambda_x : 1 \otimes x \to x

called the left unitor, a natural isomorphism

ρ x:x1x \rho_x : x \otimes 1 \to x

called the right unitor, and a natural isomorphism

B x,y:xyyx 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:

Could not include monoidal category > pentagon

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

(x1)y a x,1,y x(1y) ρ x1 y 1 xλ y xy \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:

(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) }

And lastly, we demand that

B y,xB x,y=1 xy. 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.)


The 2-category of symmetric monoidal categories

There is a strict 2-category SymmMonCatSymmMonCat with:

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:

Spectra K SymmMonCat Cat N sSet. \array{ &&&& Spectra \\ && {}^{\mathllap{K}}\nearrow && \downarrow \\ SymmMonCat &\to& Cat &\underset{N}{\to}& sSet } \,.

This construction extended to an equivalence of categories

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

between the full subcategory of the stable homotopy category Ho(Spectra)Ho(Spectra) on the connective spectra and the homotopy category of SymmMonCatSymmMonCat, 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 SymmMonCatSymmMonCat cannot carry a model category structure because it does not have all colimits. In some sense the “colimit completion” of SymmMonCatSymmMonCat 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 kk-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 k3k \geq 3

Details are in examples 1.2.3 and 1.2.4 of

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

Grothendieck ring

The Grothendieck group of a monoidal category naturally has the structure of a monoid, of an abelian monoidal category that of a ring, of an abelian braided monoidal category that of a commutative ring and, finally, of an abelian symmetric monoidal category that of a Lambda-ring. See there for more.

Internal logic

The internal logic of (closed) symmetric monoidal categories is called linear logic. This notably contains quantum logic.


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

  • The category of 2\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 vv and ww are passed past each other : vwwvv \otimes w \mapsto - w \otimes v. This non-trivial symmetric monoidal structure on Vect[ 2]Vect[\mathbb{Z}_2] defines the symmetric monoidal category of super vector spaces.


  • Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, chapter 2 of Tensor categories, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 (pdf)

A survey of definitions of symmetric monoidal categories, symmetric monoidal functors and symmetric monoidal natural transformations, is also in

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

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

Revised on October 11, 2017 12:09:04 by Mike Shulman (