# Symmetric monoidal category

In category theory, a branch of mathematics, a **symmetric monoidal category** is a braided monoidal category that is maximally symmetric. That is, the braiding operator obeys an additional identity: .

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

## Definition

A symmetric monoidal category is a monoidal category (*C*, ⊗) 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

The prototypical example is the category of vector spaces. 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, a 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. However, this category is only symmetric monoidal if*R*is commutative. - The dagger symmetric monoidal categories are symmetric monodal categories with an addtional dagger structure.

A **cosmos** is a complete cocomplete closed symmetric monoidal category.

## References

- ↑ R.W. Thomason, "Symmetric Monoidal Categories Model all Connective Spectra",
*Theory and Applications of Categories*,**Vol. 1**, No. 5, 1995, pp. 78– 118.

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