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.

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

Definition

Definition

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

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

satisfies the condition:

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

for all objects $x, y$

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 $M$ equipped with a functor

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

called the tensor product, an object

$$1 \in M$$

called the unit object, a natural isomorphism

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

called the associator, a natural isomorphism

$$\lambda_x : 1 \otimes x \to x$$

called the left unitor, a natural isomorphism

$$\rho_x : x \otimes 1 \to x$$

called the right unitor, and a natural isomorphism

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

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:

$$\array{ & (x \otimes 1) \otimes y &\stackrel{a_{x,1,y}}{\longrightarrow} & x \otimes (1 \otimes y) \\ & {}_{\rho_x \otimes 1_y}\searrow && \swarrow_{1_x \otimes \lambda_y} & \\ && x \otimes y && }$$

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

$$\array{ (x \otimes y) \otimes z &\stackrel{a_{x,y,z}}{\to}& x \otimes (y \otimes z) &\stackrel{B_{x,y \otimes z}}{\to}& (y \otimes z) \otimes x \\ \downarrow^{B_{x,y}\otimes 1_z} &&&& \downarrow^{a_{y,z,x}} \\ (y \otimes x) \otimes z &\stackrel{a_{y,x,z}}{\to}& y \otimes (x \otimes z) &\stackrel{1_y \otimes B_{x,z}}{\to}& y \otimes (z \otimes x) }$$

And lastly, we demand that

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

(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 $SymmMonCat$ with:

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

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:

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

This construction extended to an equivalence of categories

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

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

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

Relation to $\Gamma$-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) $\Gamma$-category is a functor from $\Gamma^{op}$ to $Cat$, where $\Gamma^{op}$ is a skeletal category of finite based sets and based maps. The category of $\Gamma$-categories and natural transformations, denoted by $\Gamma$$Cat$, is a symmetric monoidal closed category under the Day convolution product. The aforementioned symmetric monoidal closed model category is constructed in Sharma as a left-Bousfield localization of the projective model category structure on $\Gamma$$Cat$. A $\Gamma$-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 $k$-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 $k \geq 3$

Details are in examples 1.2.3 and 1.2.4 of

• Jacob Lurie, $\mathbb{E}[k]$-Algebras

Tannaka duality

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebra	category of modules
$A$	$Mod_A$
$R$-algebra	$Mod_R$-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
supercommutative Hopf algebra (supergroup)	rigid symmetric monoidal category with fiber functor and Schur smallness
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
$A$	$Mod_A$
$R$-2-algebra	$Mod_R$-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
$A$	$Mod_A$
$R$-3-algebra	$Mod_R$-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.

Permutations

For every $n \ge 2$ and $\sigma \in \mathfrak{S}_{n}$, there is a natural transformation

$$\sigma_{A_{1},...,A_{n}}: A_{1} \otimes ... \otimes A_{n} \rightarrow A_{\sigma^{-1}(1)} \otimes ... \otimes A_{\sigma^{-1}(n)}$$

that is defined by the decomposition of a permutation in a product of adjacent transpositions and generalizes the braiding

$$B:A_{1} \otimes A_{2} \rightarrow A_{2} \otimes A_{1}$$

We recall that an adjacent transposition $[1,n] \rightarrow [1,n]$ where $n \ge 2$ is a permutation $t:[1,n] \rightarrow [1,n]$ such that there exists a $1 \le i \le n-1$ such that:

$$t(1)=1,..., t(i-1)=i-1, t(i) = i+1, t(i+1) = i, t(i+2)=i+2,.... ,t(n)=n$$

This $i$ is then unique, and for every $1 \le i \le n-1$, we note $t_{i}$ the associated adjacent transposition.

For $n \ge 2$, every permutation $\sigma \in \mathfrak{S}_{n}$ admits at least a decomposition under the form $\sigma = t_{i_{1}}; ...;t_{i_{q}}$ where $q \ge 0$, and $1 \le i_{1},....,i_{q} \le n-1$.

In every symmetric monoidal category, for $n \ge 2$ and $1 \le i \le n-1$, we have a natural transformation

$$t_{i}:A_{1} \otimes .... \otimes A_{n} \rightarrow A_{1} \otimes ... \otimes A_{i-1} \otimes A_{i+1} \otimes A_{i} \otimes A_{i+2} \otimes .... \otimes A_{n}$$

defined by:

$$t_{i} = 1_{A_{1}} \otimes ... \otimes 1_{A_{i-1}} \otimes \B_{A_{i},A_{i+1}} \otimes 1_{A_{i+2}} \otimes ... \otimes 1_{A_{n}}$$

Given a $\sigma \in \mathfrak{S}_{n}$, the associated natural transformation is then defined as $t_{i_{1}};....;t_{i_{q}}$ for any such decomposition of $\sigma$. The fact that the result doesn’t depend on the particular decomposition is a consequence of the coherence theorem for symmetric monoidal categories.

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.

• The monoidal category of graded modules over a commutative ring (with the usual tensor product of graded modules) can be made into a braided monoidal category with the braiding

$$\array{ V \otimes W &\longrightarrow& W \otimes V \\ x \otimes y &\mapsto& y \otimes x } \,.$$

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

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

Related concepts

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

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

  • permutative category, bipermutative category

  • symmetric monoidal dagger category

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

• symmetric monoidal closed category

• equivariant symmetric monoidal category

• symmetric bicategory

• symmetric monoidal groupoid

  • symmetric 2-group, abelian ∞-group

• supply in a monoidal category

• commutative monoidal category

References

Original references:

• Saunders Mac Lane, Natural Associativity and Commutativity , Rice University Studies 49 (1963) 28-46 [rice:1911/62865]

• Jean Bénabou, Algèbre élémentaire dans les catégories (1964), C. R. Acad. Sci. Paris 258 (1964) 771-774 [ark:12148/bpt6k40102/f817]

• Samuel Eilenberg, G. Max Kelly, Part III of: Closed Categories, S. Eilenberg, D. K. Harrison, S. MacLane, H. Röhrl (eds.): Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) 421-562 [doi:10.1007/978-3-642-99902-4]

Textbook accounts:

• Francis Borceux, Section 6.1 of: Handbook of Categorical Algebra Vol. 2: Categories and Structures $[$doi:10.1017/CBO9780511525865$]$, Encyclopedia of Mathematics and its Applications 50, Cambridge University Press (1994)

• Saunders MacLane, Ch. XI of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

• Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, Tensor Categories, AMS Mathematical Surveys and Monographs 205 (2015) $[$ISBN:978-1-4704-3441-0, pdf$]$

  (focused on tensor categories)

Exposition of basics of monoidal categories and categorical algebra:

• geometry of physics – categories and toposes, Section 2: Basic notions of categorical algebra

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

• 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)

• Brendan Fong and David I, Spivak, Supplying bells and whistles in symmetric monoidal categories, arxiv, 2019

• Daniel Schäppi, Ind-abelian categories and quasi-coherent sheaves, arxiv, 2014

• Amit Sharma, Symmetric monoidal categories and $\Gamma$-categories Theory and applications of Categories, Vol. 35, No. 14, (2020) pp. 417-512

• Stefano Kasangian and Fabio Rossi. Some remarks on symmetry for a monoidal category. Bulletin of the Australian Mathematical Society 23.2 (1981): 209-214.

• J. M. Egger. On involutive monoidal categories. Theory and Applications of Categories 25.14 (2011): 368-393.