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.

1 Formal definition
2 Yoneda's lemma
3 Other properties
4 See also
5 Notes
6 References

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

Hom(A,–) : C → Set	Hom(–,B) : C → Set
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 $\definecolor{gray}{RGB}{249,249,249}\pagecolor{gray} g \mapsto f\circ g$ 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 $\definecolor{gray}{RGB}{249,249,249}\pagecolor{gray} g \mapsto g\circ h$ for each g in Hom(Y, B).

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 → Set

where C^op is the opposite category to C.

Yoneda's lemma

Main article: Yoneda 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^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.^[1]

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.

Notes

^ Jacobson (2009), p. 149, Prop. 3.9.

References

Mac Lane, Saunders (September 1998). Categories for the Working Mathematician (Second ed.). Springer. ISBN 0-387-98403-8.
Goldblatt, Robert (2006) [1984]. Topoi, the Categorial Analysis of Logic (Revised ed.). Dover Publications. ISBN 978-0486450261. http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Gold010&id=3. Retrieved 2009-11-25.
Jacobson, Nathan (2009). Basic algebra. 2 (2nd ed.). Dover. ISBN 978-0-486-47187-7.