Functor 

LYAHFGG:

And now, we’re going to take a look at the Functor typeclass, which is basically for things that can be mapped over.

Like the book let’s look how it’s implemented:

trait Functor[F[_]]  { self =>
  /** Lift `f` into `F` and apply to `F[A]`. */
  def map[A, B](fa: F[A])(f: A => B): F[B]

  ...
}

Here are the injected operators it enables:

trait FunctorOps[F[_],A] extends Ops[F[A]] {
  implicit def F: Functor[F]
  ////
  import Leibniz.===

  final def map[B](f: A => B): F[B] = F.map(self)(f)
  
  ...
}

So this defines map method, which accepts a function A => B and returns F[B]. We are quite familiar with map method for collections:

scala> List(1, 2, 3) map {_ + 1}
res15: List[Int] = List(2, 3, 4)

Scalaz defines Functor instances for Tuples.

scala> (1, 2, 3) map {_ + 1}
res28: (Int, Int, Int) = (1,2,4)

Function as Functors 

Scalaz also defines Functor instance for Function1.

scala> ((x: Int) => x + 1) map {_ * 7}
res30: Int => Int = <function1>

scala> res30(3)
res31: Int = 28

This is interesting. Basically map gives us a way to compose functions, except the order is in reverse from f compose g. No wonder Scalaz provides ∘ as an alias of map. Another way of looking at Function1 is that it’s an infinite map from the domain to the range. Now let’s skip the input and output stuff and go to Functors, Applicative Functors and Monoids.

How are functions functors? …

What does the type fmap :: (a -> b) -> (r -> a) -> (r -> b) for this instance tell us? Well, we see that it takes a function from a to b and a function from r to a and returns a function from r to b. Does this remind you of anything? Yes! Function composition!

Oh man, LYAHFGG came to the same conclusion as I did about the function composition. But wait..

ghci> fmap (*3) (+100) 1
303
ghci> (*3) . (+100) $ 1  
303 

In Haskell, the fmap seems to be working as the same order as f compose g. Let’s check in Scala using the same numbers:

scala> (((_: Int) * 3) map {_ + 100}) (1)
res40: Int = 103

Something is not right. Let’s compare the declaration of fmap and Scalaz’s map operator:

fmap :: (a -> b) -> f a -> f b

and here’s Scalaz:

final def map[B](f: A => B): F[B] = F.map(self)(f)

So the order is completely different. Since map here’s an injected method of F[A], the data structure to be mapped over comes first, then the function comes next. Let’s see List:

ghci> fmap (*3) [1, 2, 3]
[3,6,9]

and

scala> List(1, 2, 3) map {3*}
res41: List[Int] = List(3, 6, 9)

The order is reversed here too.

[We can think of fmap as] a function that takes a function and returns a new function that’s just like the old one, only it takes a functor as a parameter and returns a functor as the result. It takes an a -> b function and returns a function f a -> f b. This is called lifting a function.

ghci> :t fmap (*2)  
fmap (*2) :: (Num a, Functor f) => f a -> f a  
ghci> :t fmap (replicate 3)  
fmap (replicate 3) :: (Functor f) => f a -> f [a]  

Are we going to miss out on this lifting goodness? There are several neat functions under Functor typeclass. One of them is called lift:

scala> Functor[List].lift {(_: Int) * 2}
res45: List[Int] => List[Int] = <function1>

scala> res45(List(3))
res47: List[Int] = List(6)

Functor also enables some operators that overrides the values in the data structure like >|, as, fpair, strengthL, strengthR, and void:

scala> List(1, 2, 3) >| "x"
res47: List[String] = List(x, x, x)

scala> List(1, 2, 3) as "x"
res48: List[String] = List(x, x, x)

scala> List(1, 2, 3).fpair
res49: List[(Int, Int)] = List((1,1), (2,2), (3,3))

scala> List(1, 2, 3).strengthL("x")
res50: List[(String, Int)] = List((x,1), (x,2), (x,3))

scala> List(1, 2, 3).strengthR("x")
res51: List[(Int, String)] = List((1,x), (2,x), (3,x))

scala> List(1, 2, 3).void
res52: List[Unit] = List((), (), ())