Hom functor

In

. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics.

Formal definition

Let C be a

proper classes

).

For all objects A and B in C we define two functors to the category of sets as follows:

Hom(A, –) : C → Set	Hom(–, B) : C → Set[1]
This is a covariant functor given by: Hom(A, –) maps each object X in C to the set of morphisms, Hom(A, X) Hom(A, –) maps each morphism f : X → Y to the function Hom(A, f) : Hom(A, X) → Hom(A, Y) given by ${\displaystyle g\mapsto f\circ g}$ for each g in Hom(A, X).	This is a contravariant functor given by: Hom(–, B) maps each object X in C to the set of morphisms, Hom(X, B) Hom(–, B) maps each morphism h : X → Y to the function Hom(h, B) : Hom(Y, B) → Hom(X, B) given by ${\displaystyle g\mapsto g\circ h}$ for each g in Hom(Y, B).

The functor Hom(–, B) is also called the

of the object B.

Note that fixing the first argument of Hom naturally gives rise to a covariant functor and fixing the second argument naturally gives a contravariant functor. This is an artifact of the way in which one must compose the morphisms.

The pair of functors Hom(A, –) and Hom(–, B) are related in a natural manner. For any pair of morphisms f : B → B′ and h : A′ → A the following diagram commutes:

Both paths send g : A → B to f ∘ g ∘ h : A′ → B′.

The commutativity of the above diagram implies that Hom(–, –) is a

from C × C to Set which is contravariant in the first argument and covariant in the second. Equivalently, we may say that Hom(–, –) is a bifunctor

Hom(–, –) : C^op × C → Set

where C^op is the opposite category to C. The notation Hom_C(–, –) is sometimes used for Hom(–, –) in order to emphasize the category forming the domain.

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′)

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

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

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

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

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

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

where I is the

, namely, that

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

where ${\displaystyle \otimes }$ is a

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

.

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

(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 ${\displaystyle \operatorname {id} _{C}\colon C\nrightarrow C}$.

The internal hom functor preserves limits; that is, ${\displaystyle \operatorname {hom} (X,-)\colon C\to C}$ sends limits to limits, while ${\displaystyle \operatorname {hom} (-,X)\colon C^{\text{op}}\to C}$ sends limits in ${\displaystyle C^{\text{op}}}$, that is

in ${\displaystyle C}$, into limits. In a certain sense, this can be taken as the definition of a limit or colimit.

The

.

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.^[2]

Let R be a

functor – ${\displaystyle \otimes }$_R M: Ab → Mod-R.

See also

• Ext functor
• Functor category
• Representable functor

Notes

References

External links

• Hom functor at the nLab
• Internal Hom at the nLab