Closed monoidal category

Summary

A closed monoidal category is a monoidal category in which morphisms between two objects form again an object, the internal hom. The internal hom $[B, C]$ is characterised by a currying bijection between morphisms $A \otimes B \to C$ and morphisms $A \to [B, C]$. Cartesian closed categories are the case where $\otimes$ is the product. Vector spaces, abelian groups and modules over a commutative ring, with the tensor product, are closed monoidal.