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 wherefmap :: (a -> b) -> f a -> f b
This means that all data structures that are Functors must implement the
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
fmap is a Functor of type
a. A Functor of type
b is returned.
Lets look at an example: The list data structure.
Lists are homogeneous containers which hold 0 or more items. How can we
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  wherefmap _  = fmap f (x:xs) = f x : fmap f xs
Awesome! A short recursive function which applies
x, and appends the
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.
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 wherefmap _ Nil = Nilfmap 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 tt :: Num a => Tree aghci> fmap (*10) tNode 20 (Node 10 Nil Nil) (Node 60 Nil Nil)ghci> fmap show tNode "2" (Node "1" Nil Nil) (Node "6" Nil Nil)
Another less intuitive instance of the Functor typeclass is the
Maybe can be
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
instance Functor Maybe wherefmap _ Nothing = Nothingfmap 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 asafeHead  = NothingsafeHead (x:xs) = Just x
If the list is empty we return
Nothing, otherwise we return
ghci> safeHead [1,2,3]Just 1ghci> safeHead 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
doubleHead :: [Int] -> Maybe IntdoubleHead l =case safeHead l ofNothing -> NothingJust x -> Just (x^2)
Maybe is an instance of the Functor typeclass we can instead use
ghci> fmap (^2) $ safeHead [2,3,4]Just 4ghci> fmap (^2) $ safeHead Nothing
Haskell provides a nice infix operator for Functors. This is the
function, synonymous to
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)ghci> (^2) <$> safeHead [2,3,4]Just 4