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.

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:

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 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 : A′ → B′.

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 bifunctor