Closed monoidal category

In mathematics, especially in category theory, a closed monoidal category (or a monoidal closed category) is a category that is both a monoidal category and a closed category in such a way that the structures are compatible.

A classic example is the category of sets, Set, where the monoidal product of sets ${\displaystyle A}$ and ${\displaystyle B}$ is the usual cartesian product ${\displaystyle A\times B}$, and the internal Hom ${\displaystyle B^{A}}$ is the set of functions from ${\displaystyle A}$ to ${\displaystyle B}$. A non-cartesian example is the category of vector spaces, K-Vect, over a field ${\displaystyle K}$. Here the monoidal product is the usual tensor product of vector spaces, and the internal Hom is the vector space of linear maps from one vector space to another.

The internal language of closed symmetric monoidal categories is linear logic and the type system is the linear type system. Many examples of closed monoidal categories are symmetric. However, this need not always be the case, as non-symmetric monoidal categories can be encountered in category-theoretic formulations of linguistics; roughly speaking, this is because word-order in natural language matters.