HomSetXY

HomSetXY is a common notation in category theory for the hom-set between two objects X and Y in a category C, usually written Hom_C(X,Y). The elements of Hom_C(X,Y) are the morphisms with domain X and codomain Y, i.e., arrows f: X -> Y.

These hom-sets vary functorially in both variables. The collection of all hom-sets forms a bifunctor Hom: C^op × C -> Set, where for a morphism g: X' -> X and h: Y -> Y' the induced map sends f: X -> Y to h ∘ f ∘ g: X' -> Y'. In particular, fixing X yields the covariant functor Hom_C(X,-): C -> Set, and fixing Y yields the contravariant functor Hom_C(-,Y): C^op -> Set.

Examples: In the category Set, Hom_Set(X,Y) is the set of all functions from X to Y. In Ab, Hom_Ab(X,Y) consists of group homomorphisms between abelian groups X and Y. In Top, Hom_Top(X,Y) is the set of continuous maps from X to Y.

Yoneda lemma: Natural transformations from Hom_C(-, X) to any F: C^op -> Set correspond bijectively to elements of F(X). This establishes representable functors and the Yoneda embedding as central tools in category theory, linking objects to the sets of morphisms they admit.

Size considerations: In large or locally small categories, Hom-sets may be proper classes rather than sets. For standard small categories, however, each Hom_C(X,Y) is a set, making the bifunctor well-behaved. See also representable functors, Yoneda lemma, and enriched categories.