Horizontal & Vertical Compositions of Natural Transformations

ncompositions.scala

  trait Functor[F[_]] {
    def map[A, B](as: F[A])(f: A => B): F[B]
  }
  
  object Functor {
    def apply[F[_]](implicit e: Functor[F]): Functor[F] = e
  }
  
  trait ~>[F[_], G[_]] {
    def apply[A](x: F[A]): G[A]
  }
  
  def comp[A[_], B[_], C[_]](g: B ~> C)(f: A ~> B): A ~> C =
    new (A ~> C) {
      def apply[X](a: A[X]) = g(f(a))
    }
  
  /** Vertical composition of natural transformations */
  def vert[F[_]: Functor, F1[_]: Functor, G[_]: Functor, G1[_]: Functor](a: F1 ~> G1)(b: F ~> G): ~>[({ type l[x] = F1[F[x]] })#l, ({ type l[x] = G1[G[x]] })#l] =
    new (({ type l[x] = F1[F[x]] })#l ~> ({ type l[x] = G1[G[x]] })#l) {
      def apply[A](x: F1[F[A]]) =
        a(Functor[F1].map(x)(b(_)))
    }

  /** Horizontal composition of natural transformations */
  def horiz[F[_]: Functor, F1[_]: Functor, G[_]: Functor, G1[_]: Functor](a: F1 ~> G1)(b: F ~> G): ~>[({ type l[x] = F1[F[x]] })#l, ({ type l[x] = G1[G[x]] })#l] =
    new (({ type l[x] = F1[F[x]] })#l ~> ({ type l[x] = G1[G[x]] })#l) {
      def apply[A](x: F1[F[A]]): G1[G[A]] =
        Functor[G1].map(a(x))(b(_))
    }