Hom functor

Yoneda's lemma

Referring to the above commutative diagram, one observes that every morphism

h : A′ → A

gives rise to a natural transformation

Hom(h, –) : Hom(A, –) → Hom(A′, –)

and every morphism

f : B → B′

gives rise to a natural transformation

Hom(–, f) : Hom(–, B) → Hom(–, B′)

Yoneda's lemma implies that every natural transformation between Hom functors is of this form. In other words, the Hom functors give rise to a full and faithful embedding of the category C into the functor category Set^Cop (covariant or contravariant depending on which Hom functor is used).

Internal Hom functor

Some categories may possess a functor that behaves like a Hom functor, but takes values in the category C itself, rather than Set. Such a functor is referred to as the internal Hom functor, and is often written as

${\displaystyle \left[--\right]:C^{\text{op}}\times C\to C}$

to emphasize its product-like nature, or as

${\displaystyle \Rightarrow :C^{\text{op}}\times C\to C}$

to emphasize its functorial nature, or sometimes merely in lower-case:

${\displaystyle \mathrm{hom}(-,-):C^{\text{op}}\times C\to C.}$ For examples, see Category of relations.

Categories that possess an internal Hom functor are referred to as closed categories. One has that

${\displaystyle \mathrm{Hom}(I,\mathrm{hom}(-,-))\simeq \mathrm{Hom}(-,-)}$,

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

${\displaystyle \mathrm{Hom}(X,Y\Rightarrow Z)\simeq \mathrm{Hom}(X\otimes Y,Z)}$

where ${\displaystyle \otimes }$ 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 ${\displaystyle Y\Rightarrow Z}$ is called the internal Hom. When ${\displaystyle \otimes }$ is the Cartesian product ${\displaystyle \times }$, the object ${\displaystyle Y\Rightarrow Z}$ is called the exponential object, and is often written as ${\displaystyle Z^Y}$.

Internal Homs, when chained together, form a language, called the internal language of the category. The most famous of these are simply typed lambda calculus, which is the internal language of Cartesian closed categories, and the linear type system, which is the internal language of closed symmetric monoidal categories.

Properties

Note that a functor of the form

Hom(–, A) : C^op → Set

is a presheaf; likewise, Hom(A, –) is a copresheaf.

A functor F : C → Set that is naturally isomorphic to Hom(A, –) for some A in C is called a representable functor (or representable copresheaf); likewise, a contravariant functor equivalent to Hom(–, A) might be called corepresentable.

Note that Hom(–, –) : C^op × C → Set is a profunctor, and, specifically, it is the identity profunctor ${\displaystyle {\mathrm{id}}_C:C\nrightarrow C}$.

The internal hom functor preserves limits; that is, ${\displaystyle \mathrm{hom}(X,-):C\to C}$ sends limits to limits, while ${\displaystyle \mathrm{hom}(-,X):C^{\text{op}}\to C}$ sends limits in ${\displaystyle C^{\text{op}}}$, that is colimits in ${\displaystyle C}$, into limits. In a certain sense, this can be taken as the definition of a limit or colimit.

The endofunctor Hom(E, –) : Set → Set can be given the structure of a monad; this monad is called the environment (or reader) monad.

Other properties

If A is an abelian category and A is an object of A, then Hom_A(A, –) is a covariant left-exact functor from A to the category Ab of abelian groups. It is exact if and only if A is projective.

Let R be a ring and M a left R-module. The functor Hom_R(M, –): Mod-R → Ab is adjoint to the tensor product functor – ${\displaystyle \otimes }$_R M: Ab → Mod-R.