Functors

A functor is a map between categories that preserves their structure. Formally, a functor F from a category C to a category D consists of an object map assigning to each object X in C an object F(X) in D, and a morphism map assigning to each morphism f: X → Y in C a morphism F(f): F(X) → F(Y) in D. These maps satisfy two laws: F(id_X) = id_{F(X)} for every object X, and F(g ∘ f) = F(g) ∘ F(f) for all composable morphisms f and g in C. When the direction of arrows is preserved, the functor is called covariant; if arrows are reversed, the functor is contravariant, meaning it maps a morphism f: X → Y in C to a morphism F(f): F(Y) → F(X) in D.

Examples illustrate the concept. The identity functor Id_C: C → C assigns each object and morphism to itself. The forgetful functor U: Grp → Set sends a group to its underlying set and a group homomorphism to its underlying function. The power set functor P: Set → Set maps a set S to its set of subsets and a function f: S → T to the image map P(f): P(S) → P(T). The Hom functor, for a fixed object A in C, yields Hom(A, −): C → Set or Hom(−, A): C^op → Set, assigning to each object X the set of morphisms from A to X (or from X to A).

In category theory, functors form the morphisms between categories, and natural transformations provide a means of comparing functors. Functors are central for transferring structures and theorems across different mathematical contexts and for organizing mathematical theories in a unified framework.