In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" ${\displaystyle \otimes }$ is defined) such that the tensor product is symmetric (i.e. ${\displaystyle A\otimes B}$ is, in a certain strict sense, naturally isomorphic to ${\displaystyle B\otimes A}$ for all objects ${\displaystyle A}$ and ${\displaystyle B}$ 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.

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 ${\displaystyle s_{AB}:A\otimes B\to B\otimes A}$ 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.

Some examples and non-examples of symmetric monoidal categories:

The classifying space (geometric realization of the nerve) of a symmetric monoidal category is an ${\displaystyle E_{\infty }}$ space, so its group completion is an infinite loop space.

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.

In a symmetric monoidal category, the natural isomorphisms ${\displaystyle s_{AB}:A\otimes B\to B\otimes A}$ are their own inverses in the sense that ${\displaystyle s_{BA}\circ s_{AB}=1_{A\otimes B}}$. If we abandon this requirement (but still require that ${\displaystyle A\otimes B}$ be naturally isomorphic to ${\displaystyle B\otimes A}$), we obtain the more general notion of a braided monoidal category.