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

The classifying space (geometric realization of the nerve) of a symmetric monoidal category is an $E_\infty$ space, so its group completion 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 $s_{AB}: A \otimes B \simeq B \otimes A$ 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 additional 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.

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.