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

Properties

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.

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 A\otimes B be naturally isomorphic to B\otimes A), we obtain the more general notion of a braided monoidal category.