In mathematics, especially in category theory, a closed monoidal category (also called a monoidal closed category) is a context where it is possible both to form tensor products of objects and to form 'mapping objects'. A classic example is the category of sets, Set, where the tensor product of sets
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The 'mapping object' referred to above is also called the 'internal Hom'. 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
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
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
Biclosed and symmetric categories
Strictly speaking, we have defined a right closed monoidal category, since we required that right 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
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.