Hom Functor - Formal Definition

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
    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
    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(–,–) : Cop × C → Set

where Cop is the opposite category to C. The notation HomC(–,–) is sometimes used for Hom(–,–) in order to emphasize the category forming the domain.

Read more about this topic:  Hom Functor

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