ially in category theory, aclosed monoidal category is a context where we can take tensor products of objects and also form 'mapping objects'. A classic example is the category of sets, Set, where the tensor product of sets $A$ and $B$ is the usual cartesian product $A\; \backslash times\; B$, and the mapping object $B^A$ is the set of functions from $A$ to $B$. Another example is the category FdVect, consisting of finite-dimensional vector spaces and linear maps. Here the tensor product is the usual tensor product of vector spaces, and the mapping object is the vector space of linear maps from one vector space to another.Technically, what we have been calling a 'mapping object' is called the 'internal Hom'.

A closed monoidal category is a monoidal category $C$ such that for every object $A$ the functor given by left tensoring with $A$

$B\backslash mapsto\; A\backslash otimes\; B$

$B\backslash mapsto\; (A\; \backslash Rightarrow\; B).$

$\backslash mathbf(A\backslash otimes\; B,\; C)\backslash cong\backslash mathbf(B,A\backslash Rightarrow\; C)$

an object $A\backslash Rightarrow\; B$,

a morphism $\backslash mathrm\_$

A\otimes (A\Rightarrow B)\to B,

$f$

A\otimes X\to B

$h$

X \to A\Rightarrow B

$f\; =\; \backslash mathrm\_\backslash circ(\backslash mathrm\_A\backslash otimes\; h).$

C^ \otimes C \to C. This functor is called the internal Hom functor, and the object $A\; \backslash Rightarrow\; B$ is called the internal Hom of $A$ and $B$. Many other notations are in common use for the internal Hom. When the tensor product on C is the cartesian product, the usual notation is $B^A$.

$B\backslash mapsto\; B\backslash otimes\; A$

$B\backslash mapsto(B\backslash Leftarrow\; A)$

The monoidal category Set of sets and functions, with cartesian product as the tensor product, is a closed monoidal category. Here $A\; \backslash Rightarrow\; B$ is the set of functions from $A$ to $B$. This example is a cartesian closed category.

More generally, every cartesian closed category is a symmetric monoidal closed category, when the monoidal structure is the cartesian product structure. Here the internal hom $A\; \backslash Rightarrow\; B$ is usually written $B^A$.

The monoidal category FdVect of finite-dimensional vector spaces and linear maps, with its usual tensor product, is a closed monoidal category. Here $A\; \backslash Rightarrow\; B$ is the vector space of linear maps from $A$ to $B$. This example is a compact closed category.

More generally, every compact closed category is a symmetric monoidal closed category, in which the internal Hom functor $A\backslash Rightarrow\; B$ is given by $B\backslash otimes\; A^$

.