In category theory, a branch of mathematics, a **symmetric monoidal category** is a monoidal category (i.e. a category in which a "tensor product" is defined) such that the tensor product is symmetric (i.e. is, in a certain strict sense, naturally isomorphic to for all objects and 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 that is natural in both *A* and *B* and such that the following diagrams commute:

In the diagrams above, *a*, *l* , *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. Like before, the tensor product is just the cartesian 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**,) and (**Ste**,) of stereotype spaces over are symmetric monoidal, and moreover, (**Ste**,) is a closed symmetric monoidal category with the internal hom-functor .

## Properties

The classifying space (geometric realization of the nerve) of a symmetric monoidal category is an space, so its group completion is an infinite loop space.^{[1]}

## 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 are their *own* inverses in the sense that . If we abandon this requirement (but still require that be naturally isomorphic to ), we obtain the more general notion of a braided monoidal category.

## References

**^**Robert Wayne Thomason, "Symmetric Monoidal Categories Model all Connective Spectra",*Theory and Applications of Categories*,**Vol. 1**, No. 5, 1995, pp. 78– 118.

- Symmetric monoidal category in
*nLab* *This article incorporates material from Symmetric monoidal category on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*