symmetric monoidal category


Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



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:

Layer 1 ( w x ) ( y z ) (w\otimes x)\otimes(y\otimes z) ( ( w x ) y ) z ((w\otimes x)\otimes y)\otimes z w ( x ( y z ) ) w\otimes (x\otimes(y\otimes z)) ( w ( x y ) ) z (w\otimes (x\otimes y))\otimes z w ( ( x y ) z ) w\otimes ((x\otimes y)\otimes z) a w x , y , z a_{w\otimes x,y,z} a w , x , y z a_{w,x,y\otimes z} a w , x , y 1 z a_{w,x,y}\otimes 1_{z} 1 w a x , y , z 1_w\otimes a_{x,y,z} a w , x y , z a_{w,x\otimes y,z}

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,y1 z a y,z,x (yx)z a y,x,z y(xz) 1 yB 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 1_z} &&&& \downarrow^{a_{y,z,x}} \\ (y \otimes x) \otimes z &\stackrel{a_{y,x,z}}{\to}& y \otimes (x \otimes z) &\stackrel{1_y \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:

This 2-category has (weak) 2-biproducts given by the cartesian product of underlying categories (analogously to how Ab has biproducts given by the cartesian product of underlying sets). For a proof, see Fong-Spivak, Theorem 2.3, or for a more abstract version involving pseudomonoids Schaeppi, Appendix A.

As models for connective spectra

The group completion of 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.)

However, a subcategory of SymmMonCatSymmMonCat whose objects are Permutative categories and maps are strict symmetric monoidal functors, denoted by PermPerm has a model category structure which is transferred from the natural model category structure on CatCat, see Sharma. This model category structure is combinatorial, left-proper and a CatCat-model category structure. It is referred to as the natural model category structure on PermPerm. The coherence theorem for symmetric monoidal categories states that each symmetric monoidal category is equivalent to a permutative category.

Relation to Γ\Gamma-categories

The aforementioned natural model category of permutative categories is NOT a symmetric monoidal closed model category. This shortcoming was overcome in [Sharma] by constructing a Quillen equivalent model category which is symmetric monoidal closed. A (unnormalized) Γ\Gamma-category is a functor from Γ op\Gamma^{op} to CatCat, where Γ op\Gamma^{op} is a skeletal category of finite based sets and based maps. The category of Γ\Gamma-categories and natural transformations, denoted by Γ\GammaCatCat, is a symmetric monoidal closed category under the Day convolution product. The aforementioned symmetric monoidal closed model category is constructed in Sharma as a left-Bousfield localization of the projective model category structure on Γ\GammaCatCat. A Γ\Gamma-category is fibrant in this model category if it satisfies the Segal’s condition in which case it is referred to as a coherently commutative monoidal category. The main result of Sharma is that an unnormalized version of the classical Segal’s nerve functor is the right Quillen functor of a Quillen equivalence between the natural model category of permutative categories and the symmetric monoidal closed model category of coherently commutative monoidal categories.

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.


VW WV xy yx. \array{ V \otimes W &\longrightarrow& W \otimes V \\ x \otimes y &\mapsto& y \otimes x } \,.

The braiding xy(1) |x||y|yxx \otimes y \mapsto (-1)^{|x| |y|} y \otimes x (where |x||x| and |y||y| denote the degrees) is also commonly used.

More generally, for any invertible element uu of the base ring, there is the braiding xyu |x||y|yxx \otimes y \mapsto u^{|x| |y|} y \otimes x, and these braidings are the only possible. The resulting braided monoidal category is symmetric if and only if u 2=1u^2 = 1.


Exposition of basics of monoidal categories and categorical algebra:

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

More discussion is in