Some categories may possess a functor that behaves like a Hom functor, but takes values in the category C itself. Such a functor is referred to as the internal Hom functor, and is often written as
to emphasize its product-like nature, or as
to emphasize its functorial nature, or sometimes merely in lower-case:
Categories that possess an internal Hom functor are referred to as closed categories. The forgetful functor on such categories takes the internal Hom functor to the external Hom functor. That is,
where denotes a natural isomorphism; the isomorphism is natural in both sites. Alternately, one has that
- ,
where I is the unit object of the closed category. For the case of a closed monoidal category, this extends to the notion of currying, namely, that
where is a bifunctor, the internal product functor defining a monoidal category. The isomorphism is natural in both X and Z. In other words, in a closed monoidal category, the internal hom functor is an adjoint functor to the internal product functor. The object is called the internal Hom. When is the Cartesian product, the object is called the exponential object, and is often written as .
Read more about this topic: Hom Functor
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