# Symmetric monoidal category

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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 ${\displaystyle s_{AB}}$ obeys an additional identity: ${\displaystyle s_{BA}\circ s_{AB}=1_{A\otimes B}}$.

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 ${\displaystyle s_{AB}:A\otimes B\simeq B\otimes A}$ 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 , 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

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