Hom functor
From Exampleproblems
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 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
.
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
.
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(–,–) : Cop × C → Set
where Cop 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 SetC (covariant or contravariant depending on which Hom functor is used).
