Symmetric monoidal category

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.