Hom functor

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.

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 covariant functor:Hom("A",–) : "C" → Setto 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 $g\; mapsto\; fcirc\; g$.for each g in Hom("A", "X").

For each object "B" in "C" we define a contravariant functor:Hom(–,"B") : "C" → Setas 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 $g\; mapsto\; gcirc\; h$.

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" 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" → Setwhere "C"^op is the opposite category to "C".

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).

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_Z("M",–): Ab → Mod-"R" is right adjoint to the tensor product functor – $otimes$_R M: Mod-"R" → Ab.

ee also

* Representable functor
* Ext functor
* Currying
* Cartesian closed category