homset

In category theory, the Hom-set, denoted Hom_C(X,Y), is the set of all morphisms from an object X to an object Y in a category C. The notation Hom(X,Y) is common when the involved category is clear from context. For a fixed category, the Hom-sets vary with the pair of objects (X,Y).

The composition of morphisms endows the collection of Hom-sets with a rich structure: there is a function Hom(Y,Z) × Hom(X,Y) → Hom(X,Z) given by composition, and for each object X the identity morphism yields the identity element in Hom(X,X). These rules satisfy associativity and identity laws, reflecting the categorical structure.

From a functorial perspective, fixed Y yields the contravariant functor Hom(-,Y): C^op → Set, which assigns to each X the set Hom(X,Y). Fixed X yields the covariant functor Hom(X,-): C → Set, which assigns to each Y the set Hom(X,Y). These functors underpin the Yoneda lemma, which relates objects to the set-valued functors they represent.

Examples help illuminate the concept. In the category of sets, Hom_Set(X,Y) is the set of all functions X → Y. In the category of groups, Hom_Group(G,H) is the set of group homomorphisms from G to H. In a poset regarded as a thin category, Hom(a,b) is either a singleton if a ≤ b or empty otherwise.

Variants and generalizations include enriched category theory, where Hom-sets are replaced by hom-objects in a monoidal category, and locally small categories, where all Hom-sets are actual sets. Representable functors and the Yoneda lemma are central to studying Hom-sets and their representability. See also Yoneda lemma, representable functor, and enriched category theory.