Functors

In mathematics, Functors are a mapping between categories. They define a way to convert an object in one category to an object in another.

In programming, they are basically the same thing.

A Functor is a data structure that can be mapped over.

The term Functor is most commonly used in Haskell, but nothing about them are specific to Haskell.

The Haskell Functor type class is defined as

``````class Functor f where
fmap :: (a -> b) -> f a -> f b``````

This means that all data structures that are Functors must implement the `fmap` function. `fmap` take two parameters. `(a -> b)` is a function which takes a single parameter of type `a` and returns an item of type `b`. The second parameter to `fmap` is a Functor of type `a`. A Functor of type `b` is returned.

Lists

Lets look at an example: The list data structure.

Lists are homogeneous containers which hold 0 or more items. How can we implement a `fmap` function that allows us to map over every item in the list? In other words, we need to apply a function to every element in the list and create a new list with the result of each function application.

``````instance Functor [] where
fmap _ [] = []
fmap f (x:xs) = f x : fmap f xs``````

Awesome! A short recursive function which applies `f` to `x`, and appends the result to `fmap` applied to the tail (rest) of the list. Our base case is when we get an empty list.

We can now map over any list like so

``````ghci> fmap (^2) [1,2,3]
[1,4,9]

ghci> fmap show [4,5,6]
["4","5","6"]``````

Lists are the most intuitive example for functors, but any data structure can become one.

Binary Trees

Let's look at binary trees.

A binary tree can be either empty or of some value with subtrees on the left and right. In Haskell we would represent that as

``````data Tree a
= Nil
| Node a (Tree a) (Tree a)
deriving (Show)``````

We can make `Tree` a Functor by making it an instance of the typeclass

``````instance Functor Tree where
fmap _ Nil = Nil
fmap f (Node x left right) =
Node (f x) (fmap f left) (fmap f right)``````

All this does it apply the function `f` to every node of the tree. Now we can do things like this

``````ghci> let t = Node 2 (Node 1 Nil Nil) (Node 6 Nil Nil)

ghci> :t t
t :: Num a => Tree a

ghci> fmap (*10) t
Node 20 (Node 10 Nil Nil) (Node 60 Nil Nil)

ghci> fmap show t
Node "2" (Node "1" Nil Nil) (Node "6" Nil Nil)``````

Maybes

Another less intuitive instance of the Functor typeclass is the `Maybe` type.

A `Maybe` can be `Nothing` or `Just` a value.

When you map over a Maybe with the function `f`, it will apply `f` to the contents of the Maybe if it is `Just` a value, otherwise it just returns `Nothing`.

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

As an example, lets say we want to get the head of a list but not crash our program if the list is empty.

``````safeHead :: [a] -> Maybe a

If the list is empty we return `Nothing`, otherwise we return `Just x`.

``````ghci> safeHead [1,2,3]
Just 1

Nothing``````

Now if, for whatever reason, we want to square the head of the list, we could pattern match on the `Maybe` type to extract the wrapped value and apply the function.

``````doubleHead :: [Int] -> Maybe Int
Nothing -> Nothing
Just x -> Just (x^2)``````

But since `Maybe` is an instance of the Functor typeclass we can instead use `fmap`.

``````ghci> fmap (^2) \$ safeHead [2,3,4]
Just 4

ghci> fmap (^2) \$ safeHead []
Nothing``````

Much nicer!

Sugar

Haskell provides a nice infix operator for Functors. This is the `(<\$>)` function, synonymous to `fmap`.

``````ghci> :t (<\$>)
(<\$>) :: Functor f => (a -> b) -> f a -> f b``````

Our previous examples could have been written like

``````ghci> (^2) <\$> [1,2,3]
[1,4,9]

ghci> (*10) <\$> Node 2 (Node 1 Nil Nil) (Node 6 Nil Nil)
Node 20 (Node 10 Nil Nil) (Node 60 Nil Nil)