Maths - Functor - Code

Implementing functors in computer code.

Haskell Code

Code from here.

The 'Functor' class is used for types that can be mapped over. Instances of 'Functor' should satisfy the following laws:

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

    -- Replace all locations in the input with the same value.
    -- The default definition is @'fmap' . 'const'@, but this may be
    -- overridden with a more efficient version.
    (<$)        :: a -> f b -> f a
    (<$)        =  fmap . const

Some Instances

The instances of 'Functor' satisfy these laws.

List functor

instance Functor [] where
    fmap = map

IO functor

instance Functor ((->) r) where
    fmap = (.)

Maybe functor

instance Functor ((,) a) where
    fmap f (x,y) = (x, f y)

Scala Code

Code from here. package scalaz

Covariant function application in an environment. i.e. a covariant Functor.

All functor instances must satisfy 2 laws:

trait Functor[F[_]] extends InvariantFunctor[F] {
  def fmap[A, B](r: F[A], f: A => B): F[B]

  final def xmap[A, B](ma: F[A], f: A => B, g: B => A) = fmap(ma, f)

object Functor {
  import Scalaz._

  implicit def IdentityFunctor: Functor[Identity] = new Functor[Identity] {
    def fmap[A, B](r: Identity[A], f: A => B) = f(r.value)

  implicit def TraversableFunctor[CC[X] <: collection.TraversableLike[X, CC[X]] :
                                      CanBuildAnySelf]: Functor[CC] = new Functor[CC] {
    def fmap[A, B](r: CC[A], f: A => B) = {
      implicit val cbf = implicitly[CanBuildAnySelf[CC]].builder[A, B]
      r map f

  implicit def NonEmptyListFunctor = new Functor[NonEmptyList] {
    def fmap[A, B](r: NonEmptyList[A], f: A => B) = r map f

  implicit def ConstFunctor[BB: Monoid] = new Functor[({type λ[α]=Const[BB, α]})#λ] {
    def fmap[A, B](r: Const[BB, A], f: (A) => B) = Const(r.value)

  implicit def StateFunctor[S] = new Functor[({type λ[α]=State[S, α]})#λ] {
    def fmap[A, B](r: State[S, A], f: A => B) = r map f

Extending these examples to Natural Transformations

We can extend these examples to represent natural transformations as shown on page here.

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see also:

Correspondence about this page

Book Shop - Further reading.

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Terminology and Notation

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