Hom functor: Encyclopedia II - Hom functor - Formal definition

Hom functor - 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

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

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

where C^op is the opposite category to C.

Adapted from the Wikipedia article "Formal definition", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki