shapes

shapes.scala

   (A => B) => A    => B    // function
   (A => B) => F[A] => F[B] // Functor.fmap
  Z[A => B] => Z[A] => Z[B] // Applicative.apply
(A => M[B]) => M[A] => M[B] // Monad.bind

(A => B) apply           A: B    // Function1.apply
    F[A] fmap     (A => B): F[B] // Functor.fmap
    Z[A] <*>     Z[A => B]: Z[B] // Applicative.apply
    M[A] >>=   (A => M[B]): M[B] // Monad.bind

K[M, A, B] >=> (B => M[C]): K[M, A, C] // Kleisli compose

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

trait Applicative[A[_]] extends Functor[A] {
  def <*>[A, B](a: Z[A], f: Z[A => B]): Z[B]
}

val za: Z[A]
val zb: Z[B]

/*
   zb   <*> (za   fmap f)
   Z[B] <*> (Z[A] fmap (A => B => C))
   Z[B] <*> (Z[B => C])
   Z[C]
 */
val f: (A => B => C)
val zc: Z[C] = zb <*> (za fmap f)


/*
   Order doesn't matter!

   za   <*> (zb   fmap f2)
   Z[A] <*> (Z[B] fmap (B => A => C))
   Z[A] <*> (Z[A => C])
   Z[C]
 */
val f2: (B => A => C)
val zc: Z[C] = za <*> (zb fmap f2)

/*
 * Sources:
 * 
 * Functors and things using Scala, http://blog.tmorris.net/posts/functors-and-things-using-scala/
 * Lifting, http://blog.tmorris.net/posts/lifting/
 */

The fmap operation is sometimes called lift1 It corresponds to lifting an arity-1 function into an environment (F). See http://blog.tmorris.net/posts/lifting/

Lines 13-19 do not accurately represent the operation under examination, since they omit the initial argument (F[A]). The use of this is not constrained enough. However, they are on the right track and with some modification, can be made more accurate.