Endofunctor

An endofunctor of a category C is a functor F: C → C, i.e., a mapping that assigns to each object X in C an object F(X) in C and to each morphism f: X → Y a morphism F(f): F(X) → F(Y), preserving identities and composition.

Endofunctors are defined for any category, not just familiar ones like Set. Common examples include the identity functor 1_C, which sends every object and morphism to itself; the power set functor P on Set, which maps a set X to its power set P(X) and a function f: X → Y to P(f): P(X) → P(Y) by image; and the list functor that sends a set X to the set of finite lists of elements of X, with the action on functions applying f to each element of the list.

Endofunctors can be composed: if F and G are endofunctors on C, their composition F ∘ G is again an endofunctor C → C. Consequently, endofunctors on a category C form a category themselves, often denoted [C, C], where the objects are endofunctors and the morphisms are natural transformations between them.

Natural transformations between endofunctors provide a way to compare functors. A natural transformation η: F ⇒ G consists of a family of arrows η_X: F(X) → G(X) natural in X. Endofunctors equipped with additional structure yield important constructions: a monad is an endofunctor F together with two natural transformations unit η: Id → F and multiplication μ: F ∘ F → F satisfying associativity and unit laws; dually, a comonad uses counit and comultiplication.

Endofunctors appear throughout category theory and its applications, including algebra, topology, and computer science, where they model various constructions and computational effects.