Functor (functional programming)

In functional programming, a functor is a design pattern inspired by the definition from category theory that allows one to apply a function to values inside a generic type without changing the structure of the generic type. In Haskell this idea can be captured in a type class:

class Functor f where
  fmap :: (a -> b) -> f a -> f b

This declaration says that any type of Functor must support a method fmap, which maps a function over the element(s) of the Functor.

Functors in Haskell should also obey functor laws, which state that the mapping operation preserves the identity function and composition of functions:

fmap id = id
fmap (g . h) = (fmap g) . (fmap h)

(where . stands for function composition).

In Scala a trait can be used:

trait Functor[F[_]] {
  def map[A,B](a: F[A])(f: A => B): F[B]
}

Functors form a base for more complex abstractions like Applicative Functor, Monad, and Comonad, all of which build atop a canonical functor structure. Functors are useful in modeling functional effects by values of parameterized data types. Modifiable computations are modeled by allowing a pure function to be applied to values of the "inner" type, thus creating the new overall value which represents the modified computation (which might yet to be run).