scalaz type constructors

scalaz.scala

//
// Foldable
// - fold a data structure
// - usually, fold F[A] given a Monoid[A]
//

trait Foldable[F[_]]  { self =>
  def foldRight[A, B](fa: F[A], z: => B)(f: (A, => B) => B): B

  def foldLeft[A, B](fa: F[A], z: B)(f: (B, A) => B): B = {

  def foldMap[A,B](fa: F[A])(f: A => B)(implicit F: Monoid[B]): B

  ...
}

//
// Functor
// - map lifts a function f into functor F and applies it
//   to the value within functor F i.e. F[A]
//
trait Functor[F[_]] extends InvariantFunctor[F] { self =>
  def map[A, B](fa: F[A])(f: A => B): F[B]

  ...
}

//
// Apply
// - ap takes a functor that that has a function in it and another
//   functor and extracts that function from the first functor and
//   then maps it over the second one
//
trait Apply[F[_]] extends Functor[F] { self =>
  def ap[A, B](fa: => F[A])(f: => F[A => B]): F[B]

  ...
}

//
// Applicative
//
trait Applicative[F[_]] extends Apply[F] { self =>
  def point[A](a: => A): F[A]

  final def pure[A](a: => A): F[A] = point(a)

  ...
}