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.

Formal definition

Let C be a locally small category.
For all objects A and B in C we define two functors to the category of sets as follows:
Hom : C → SetHom : C → Set
This is a covariant functor given by:
  • Hom maps each object X in C to the set of morphisms, Hom
  • Hom maps each morphism f : X → Y to the function
  • : Hom : Hom → Hom given by
  • : for each g in Hom.
This is a contravariant functor given by:
  • Hom maps each object X in C to the set of morphisms, Hom
  • Hom maps each morphism h : X → Y to the function
  • : Hom : Hom → Hom given by
  • : for each g in Hom.
    • The functor Hom 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 and Hom 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 covariant bifunctor
    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.

    Yoneda's lemma

    Referring to the above commutative diagram, one observes that every morphism
    gives rise to a natural transformation
    and every morphism
    gives rise to a natural transformation
    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 SetCop.

    Internal Hom functor

    Some categories may possess a functor that behaves like a Hom functor, but takes values in the category C itself, rather than Set. Such a functor is referred to as the internal Hom functor, and is often written as
    to emphasize its product-like nature, or as
    to emphasize its functorial nature, or sometimes merely in lower-case:
    Categories that possess an internal Hom functor are referred to as closed categories. The forgetful functor on such categories takes the internal Hom functor to the external Hom functor. That is,
    where denotes a natural isomorphism; the isomorphism is natural in both sides. Alternately, one has that
    where I is the unit object of the closed category. For the case of a closed monoidal category, this extends to the notion of currying, namely, that
    where is a bifunctor, the internal product functor defining a monoidal category. The isomorphism is natural in both X and Z. In other words, in a closed monoidal category, the internal Hom functor is an adjoint functor to the internal product functor. The object is called the internal Hom. When is the Cartesian product, the object is called the exponential object, and is often written as.
    Internal Homs, when chained together, form a language, called the internal language of the category. The most famous of these are simply typed lambda calculus, which is the internal language of Cartesian closed categories, and the linear type system, which is the internal language of closed symmetric monoidal categories.

    Properties

    Note that a functor of the form
    is a presheaf; likewise, Hom is a copresheaf.
    A functor F : C → Set that is naturally isomorphic to Hom for some A in C, is called a representable functor ; likewise, a contravariant functor equivalent to Hom might be called corepresentable.
    Note that Hom : Cop × C → Set is a profunctor, and, specifically, it is the identity profunctor.
    The internal hom functor preserves limits; that is, sends limits to limits, while sends limits in, that is colimits, into limits. In a certain sense, this can be taken as the definition of a limit or colimit.

    Other properties

    If A is an abelian category and A is an object of A, then HomA 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.
    Let R be a ring and M a left R-module. The functor HomR: Mod-R → Ab is right adjoint to the tensor product functor - R M: Ab → Mod-R.