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 C is the cartesian product, the usual notation is and this object is called the exponential object.
Read more about Closed Monoidal Category: Biclosed and Symmetric Categories, Examples
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