hombifunctor

A hombifunctor, commonly called the Hom bifunctor, is a two-variable functor that encodes morphisms in a category. In a category C, the hombifunctor is the functor Hom_C: C^op × C → Set, which maps a pair of objects (A, B) to the hom-set Hom_C(A, B). On morphisms, given f: A' → A in C and g: B → B' in C, Hom_C sends (f, g) to the function Hom_C(A, B) → Hom_C(A', B') defined by φ ↦ g ∘ φ ∘ f. Thus it is contravariant in the first argument and covariant in the second.

More generally, a hombifunctor may refer to any bifunctor F: C^op × D → E that is contravariant in the first argument and covariant in the second. Such bifunctors are functorial in each variable and satisfy naturality with respect to morphisms in C and D.

Key properties: For fixed B, the functor Hom_C(-, B): C^op → Set is contravariant; for fixed A, the functor Hom_C(A, -): C → Set is covariant. The Hom bifunctor is representable in both variables, and through the Yoneda lemma it underpins many results about representable functors and adjunctions.

Examples: In the category Set, Hom_Set(A, B) is the set of all functions from A to B. In any category, Hom_C(A, B) generalizes the notion of function sets to morphisms. In enriched contexts, Hom_C(A, B) may carry additional structure, such as being an abelian group or an object of a monoidal category.

Usage: The concept is central to representable functors, natural transformations, and the formulation of adjunctions. The term “hombifunctor” is typically used interchangeably with the Hom bifunctor.