Concept
Foncteur Hom
In mathematics, specifically in , hom-sets (i.e. sets of morphisms between ) give rise to important functors to the . These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics. Let C be a (i.e. a 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 as follows: {| class=wikitable |- ! 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 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 |
| Hom(–, –) : Cop × C → Set |
| where Cop is the to C. The notation HomC(–, –) is sometimes used for Hom(–, –) in order to emphasize the category forming the domain. |
| Yoneda 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 implies that every natural transformation between Hom functors is of this form. |
Source officielle
À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.