A closed monoidal category is a monoidal category such that for every object the functor given by right tensoring with has a right adjoint, written This means that there exists a bijection, called 'currying', between the Hom-sets that is natural in both A and C. In a different, but common notation, one would say that the functor has a right adjoint Equivalently, a closed monoidal category is a category equipped, for every two objects A and B, with
an object,
a morphism,
satisfying the following universal property: for every morphism there exists a unique morphism such that It can be shown that this construction defines a functor. This functor is called the internal Hom functor, and the object is called the internal Hom of and. Many other notations are in common use for the internal Hom. When the tensor product on is the cartesian product, the usual notation is and this object is called the exponential object.
Biclosed and symmetric categories
Strictly speaking, we have defined a right closed monoidal category, since we required that right tensoring with any object has a right adjoint. In a left closed monoidal category, we instead demand that the functor of left tensoring with any object have a right adjoint A biclosed monoidal category is a monoidal category that is both left and right closed. A symmetric monoidal category is left closed if and only if it is right closed. Thus we may safely speak of a 'symmetric monoidal closed category' without specifying whether it is left or right closed. In fact, the same is true more generally for braided monoidal categories: since the braiding makes naturally isomorphic to, the distinction between tensoring on the left and tensoring on the right becomes immaterial, so every right closed braided monoidal category becomes left closed in a canonical way, and vice versa. We have described closed monoidal categories as monoidal categories with an extra property. One can equivalently define a closed monoidal category to be a closed category with an extra property. Namely, we can demand the existence of a tensor product that is left adjoint to the internal Hom functor. In this approach, closed monoidal categories are also called monoidal closed categories.
Examples
Every cartesian closed category is a symmetric, monoidal closed category, when the monoidal structure is the cartesian product structure. The internal Hom functor is given by the exponential object.
* In particular, the category of sets, Set, is a symmetric, closed monoidal category. Here the internal Hom is just the set of functions from to.
* In particular, the category of vector spaces over a field is a symmetric, closed monoidal category.
* Abelian groups can be regarded as Z-modules, so the category of abelian groups is also a symmetric, closed monoidal category.
A compact closed category is a symmetric, monoidal closed category, in which the internal Hom functor is given by. The canonical example is the category of finite-dimensional vector spaces, FdVect.
Counterexamples
The category of rings is a symmetric, monoidal category under the tensor product of rings, with serving as the unit object. This category is not closed. If it were, there would be exactly one homomorphism between any pair of rings:. The same holds for the category of R-algebras over a commutative ring R.
