Symmetric monoidal category

In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" ${\displaystyle \otimes }$ is defined) such that the tensor product is symmetric (i.e. ${\displaystyle A\otimes B}$ is, in a certain strict sense, naturally isomorphic to ${\displaystyle B\otimes A}$ for all objects ${\displaystyle A}$ and ${\displaystyle B}$ of the category). One of the prototypical examples of a symmetric monoidal category is the category of vector spaces over some fixed field k, using the ordinary tensor product of vector spaces.

Definition

A symmetric monoidal category is a monoidal category (C, ⊗, I) such that, for every pair A, B of objects in C, there is an isomorphism ${\displaystyle s_{AB}:A\otimes B\to B\otimes A}$ called the swap map^[1] that is natural in both A and B and such that the following diagrams commute:

• The unit coherence:

  

• The associativity coherence:

  

• The inverse law:

  

In the diagrams above, a, l, and r are the associativity isomorphism, the left unit isomorphism, and the right unit isomorphism respectively.

Examples

Some examples and non-examples of symmetric monoidal categories:

• The category of sets. The tensor product is the set theoretic cartesian product, and any singleton can be fixed as the unit object.
• The category of groups. The tensor product is the direct product of groups, and the trivial group is the unit object.
• More generally, any category with finite products, that is, a cartesian monoidal category, is symmetric monoidal. The tensor product is the direct product of objects, and any terminal object (empty product) is the unit object.
• The category of bimodules over a ring R is monoidal (using the ordinary tensor product of modules), but not necessarily symmetric. If R is commutative, the category of left R-modules is symmetric monoidal. The latter example class includes the category of all vector spaces over a given field.
• Given a field k and a group (or a Lie algebra over k), the category of all k-linear representations of the group (or of the Lie algebra) is a symmetric monoidal category. Here the standard tensor product of representations is used.
• The categories (Ste,${\displaystyle \circledast }$) and (Ste,${\displaystyle \odot }$) of stereotype spaces over ${\displaystyle {\mathbb {C} }}$ are symmetric monoidal, and moreover, (Ste,${\displaystyle \circledast }$) is a closed symmetric monoidal category with the internal hom-functor ${\displaystyle \oslash }$.

Properties

The classifying space (geometric realization of the nerve) of a symmetric monoidal category is an ${\displaystyle E_{\infty }}$ space, so its group completion is an infinite loop space.^[2]

Specializations

A dagger symmetric monoidal category is a symmetric monoidal category with a compatible dagger structure.

A cosmos is a complete cocomplete closed symmetric monoidal category.

Generalizations

In a symmetric monoidal category, the natural isomorphisms ${\displaystyle s_{AB}:A\otimes B\to B\otimes A}$ are their own inverses in the sense that ${\displaystyle s_{BA}\circ s_{AB}=1_{A\otimes B}}$. If we abandon this requirement (but still require that ${\displaystyle A\otimes B}$ be naturally isomorphic to ${\displaystyle B\otimes A}$), we obtain the more general notion of a braided monoidal category.