Endofunctors

An endofunctor on a category C is a functor F: C -> C, meaning it assigns to every object X in C an object F(X) in C and to every morphism f: X -> Y a morphism F(f): F(X) -> F(Y), preserving identities and composition. Endofunctors are elements of the category [C, C], whose objects are functors from C to C and whose morphisms are natural transformations between them. The identity functor id_C and the composition of endofunctors are basic examples, and the composite of two endofunctors is itself an endofunctor.

Examples include endofunctors on Set such as the identity functor, the list functor X |-> X*, the option/maybe functor X |-> 1 + X, and the power-set functor X |-> P(X). Similar constructions exist on other categories, such as the tree or stream functors on appropriate categories of data types.

Endofunctors are central to several theories in category theory and its applications. A monad is an endofunctor equipped with two natural transformations that satisfy coherence laws, providing a framework for modeling computational effects and algebraic theories; examples include the list monad and the maybe/option monad in programming. Endofunctors also generate F-algebras, consisting of an object X together with a morphism F(X) -> X, and F-coalgebras, which model state-based systems. Initial algebras and final coalgebras for an endofunctor yield canonical solutions to recursive definitions and coinductive specifications, respectively. Endofunctors thus serve as a versatile tool for encoding data types, processes, and iterative constructions across mathematics and computer science.