Hom functor

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In mathematics, specifically in category theory, Hom-sets, i.e. sets of morphisms between objects, give rise to important functors to the category of sets. These functors are called Hom-functors and have numerous applications in category theory and other branches of mathematics.

[edit] Formal definition

Let C be a locally small category (i.e. a category for which Hom-classes are actually sets and not proper classes). For all objects A in C we define a functor

Hom(A,–) : C → Set

to the category of sets as follows:

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 $g \mapsto f\circ g$ .

For each object B in C we define a contravariant functor

Hom(–,B) : C → Set

as follows:

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 $g \mapsto g\circ h$ .

The functor Hom(–,B) is also called the functor of points 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 obviously related in a natural manner. For any pair of morphisms f : B → B′ and h : A′ → A and the following diagram commutes:

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

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

Hom(–,–) : C^op × C → Set

where C^op is the opposite category to C.

[edit] Yoneda's lemma

Main article: 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 asserts 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^C (covariant or contravariant depending on which Hom functor is used).