Question

Im trying to understand what a functor is, i found this tutorial/example:

http://en.wikibooks.org/wiki/Haskell/Solutions/Applicative_Functors

 data Tree a = Node a [Tree a]

The functor for the above type being:

 instance Functor Tree where
   fmap f (Node a ts) = Node (f a) (map (fmap f) ts)

could someone help explain what exactly they have done and why they have done it? My understanding is that a functor allows you to iterate over a data type. I cant seem to understand the syntax used though?

Answer 1

A Functor is useful for mapping between two data representations. Sometimes that might resemble iteration, sometimes not. Having this common Functor typeclass allows us to ignore the actual structure of the data type (Maybe, List, Tree) and focus only on the data they contain. The author of that data type should know how that data structure might be traversed/iterated so he should provide the implementation for that (in the form of a Functor instance of that data type). All we have to provide is that function f which takes an a and maps it to a b. For example:

import Data.Char (toLower)

data Tree a = Node a [Tree a]
  deriving Show

instance Functor Tree where
  fmap f (Node a ts) = Node (f a) (map (fmap f) ts)

main :: IO ()
main = do print (toLower `fmap` (Node 'F' [])) -- Node 'f' []
          print (toLower `fmap` (Just 'F'))    -- Just 'f'
          print (toLower `fmap` "FOO")         -- "foo"

We were able to lowercase those chars using the same code toLower combined with fmap.

So, what you should do when defining that Functor instance is to extract inner data using pattern matching, and the apply the received callback function f to each of these results.

An infix synonym for fmap can be found in the Control.Applicative module, called <$>.

main :: IO ()
main = do print (toLower <$> (Node 'F' [])) -- Node 'f' []
          print (toLower <$> (Just 'F'))    -- Just 'f'
          print (toLower <$> "FOO")         -- "foo"