Functor

A functor is an abstract concept, so we will begin with some examples.

In order to have an example to work with, define the function:

f :: Double -> Double
f x = x*x

If you want to apply f to a single number, no problem.

ghci> x = 10
ghci> f x
100

What if you want to apply f to a bunch of numbers?

ghci> xs = [10,20,30]

You probably know about map.

ghci> map f xs
[100,400,900]

The function fmap does the same thing in this case.

ghci> fmap f xs
[100,400,900]

What would it mean to apply f to a Maybe Double? With an input of Just 10, it would make the most sense for f to output Just 100. With an output of Nothing, the sensible output (from any function) would be Nothing.

That is what fmap is made to do.

ghci> mx = Just 10
ghci> fmap f mx
Just 100
ghci> nx = Nothing
ghci> fmap f nx
Nothing

Haskell allows you create an fmap function for your own data types by making them instances of the Functor class. The definition of fmap for Maybe looks like this:

instance Functor Maybe where
    fmap f (Just x) = Just (f x)
    fmap f (Nothing) = Nothing

As you can see, it specifies how fmap f acts on all of the possibilities for input.

Exercises A

1. Give two different typed examples of the Tagged type.

    data Tagged a = Tagged String a
   

2. Is there a sensible way to apply the function sn below to a Tagged String?

    sn :: String -> (Int, String)
    sn w = (100 * length w, w)
   

3. What type(s) could you use for a that would let you Tagged a

    ns :: Int -> [String]
    ns k = replicate k "Tag"
   

Can you turn this data definition into a Functor instance?

Exercises B

Use the Data Types page for the definitions used below.

Important note about functors: fmap is only supposed to operate on the last type

1. Write the instance declaration to make Triple an instance of Functor.

    data Triple a = Never | Unlikely a | Surely a
   

2. Write the instance declaration to make Twin (below) an instance of Functor.

    data TwinL a = NotPresent | TwinR a a
   

3. Make Yikes a Functor.

    data YikesL a b = YikesR a a b
   

4. A binary tree

    data Tree a = Leaf a |
                  Tree {getValue :: a,
                        getLeft  :: (Tree a),
                        getRight :: (Tree a)}
   

   1. Make an example of a tree with only one value.
   2. Make an example of a tree with three values (one on the left, one on the right).
   3. Make Tree into a Functor.