Endofunctor

definition example

An endofunctor is a Functor $F:\mathcal{C}\to \mathcal{C}$ from a category to itself. All endofunctors of $\mathcal{C}$ compose, so they form a monoid — indeed a strict Monoidal Category $[\mathcal{C},\mathcal{C}]$ under composition, whose monoids are monads.

Sources: DaoFP §8.3 (“Endofunctors”), Chapters 12–14; Kittenlab Lecture 7 (order-preserving maps $\mathbb{R}\to \mathbb{R}$ as endofunctors).

• In Haskell an endofunctor of $\mathbf{Hask}$ is a type constructor f :: Type -> Type with an fmap: List, Maybe, Identity, Const c, WithInt (DaoFP Exercise 8.3.1).
• Algebras $Fa\to a$ and coalgebras $a\to Fa$ of endofunctors define recursive data types (Initial Algebra, Terminal Coalgebra).
• Every Haskell endofunctor is strong because $\mathbf{Hask}$ is self-enriched (DaoFP §20.2).