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 A and B is the usual cartesian product A \times B, and the internal Hom B^A is the set of functions from A to B. A non-cartesian example is the category of vector spaces, K-Vect, over a field 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.

Definition

A closed monoidal category is a monoidal category \mathcal{C} such that for every object B the functor given by right tensoring with B 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 \mathcal{C} is a category equipped, for every two objects A and B, with 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 A and B. Many other notations are in common use for the internal Hom. When the tensor product on \mathcal{C} is the cartesian product, the usual notation is B^A 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 A has a right adjoint. In a left closed monoidal category, we instead demand that the functor of left tensoring with any object A 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 A \otimes B naturally isomorphic to B \otimes A, 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

Counterexamples