In this chapter, we will introduce the important
Functor type class.
In Other data structures, we saw operations that apply to all elements of some grouped value. The prime example is
map which works on lists. Another example we worked through was the following Tree datatype:
data Tree a = Leaf a | Branch (Tree a) (Tree a) deriving (Show)
The map function we wrote for Tree was:
treeMap :: (a -> b) -> Tree a -> Tree b treeMap f (Leaf x) = Leaf (f x) treeMap f (Branch left right) = Branch (treeMap f left) (treeMap f right)
As discussed before, we can conceivably define a map-style function for any arbitrary data structure.
When we first introduced
map in Lists II, we went through the process of taking a very specific function for list elements and generalizing to show how
map combines any appropriate function with all sorts of lists. Now, we will generalize still further. Instead of map-for-lists and map-for-trees and other distinct maps, how about a general concept of maps for all sorts of mappable types?
Functor is a Prelude class for types which can be mapped over. It has a single method, called
fmap. The class is defined as follows:
class Functor f where fmap :: (a -> b) -> f a -> f b
The usage of the type variable
f can look a little strange at first. Here,
f is a parametrized data type; in the signature of
a as a type parameter in one of its appearances and
b in the other. Let's consider an instance of
Functor: By replacing
Maybe we get the following signature for
fmap :: (a -> b) -> Maybe a -> Maybe b
... which fits the natural definition:
instance Functor Maybe where fmap f Nothing = Nothing fmap f (Just x) = Just (f x)
(Incidentally, this definition is in Prelude; so, we didn't really need to have implemented
maybeMap for that example in the "Other data structures" chapter.)
Functor instance for lists (also in Prelude) is simple:
instance Functor  where fmap = map
... and if we replace
 in the
fmap signature, we get the familiar type of
fmap is a generalization of
map for any parametrized data type.
Naturally, we can provide
Functor instances for our own data types. In particular,
treeMap can be promptly relocated to an instance:
instance Functor Tree where fmap f (Leaf x) = Leaf (f x) fmap f (Branch left right) = Branch (fmap f left) (fmap f right)
Here's a quick demo of
fmap in action with the instances above (to reproduce it, you only need to load the
instance declarations for
Tree; the others are already in Prelude):
*Main> fmap (2*) [1,2,3,4] [2,4,6,8] *Main> fmap (2*) (Just 1) Just 2 *Main> fmap (fmap (2*)) [Just 1, Just 2, Just 3, Nothing] [Just 2, Just 4, Just 6, Nothing] *Main> fmap (2*) (Branch (Branch (Leaf 1) (Leaf 2)) (Branch (Leaf 3) (Leaf 4))) Branch (Branch (Leaf 2) (Leaf 4)) (Branch (Leaf 6) (Leaf 8))
The functor lawsEdit
When providing a new instance of
Functor, you should ensure it satisfies the two functor laws. There is nothing mysterious about these laws; their role is to guarantee
fmap behaves sanely and actually performs a mapping operation (as opposed to some other nonsense). The first law is:
fmap id = id
id is the identity function, which returns its argument unaltered. The first law states that mapping
id over a functor must return the functor unchanged.
Next, the second law:
fmap (f . g) = fmap f . fmap g
It states that it should not matter whether we map a composed function or first map one function and then the other (assuming the application order remains the same in both cases).
What did we gain?Edit
At this point, we can ask what benefit we get from the extra layer of generalization brought by the
Functor class. There are two significant advantages:
- The availability of the
fmapmethod relieves us from having to recall, read, and write a plethora of differently named mapping methods (
weirdMap, ad infinitum). As a consequence, code becomes both cleaner and easier to understand. On spotting a use of
fmap, we instantly have a general idea of what is going on.
- Using the type class system, we can write
fmap-based algorithms which work out of the box with any functor - be it
Treeor whichever you need. Indeed, a number of useful classes in the core libraries inherit from
Type classes make it possible to create general solutions to whole categories of problems. Depending on what you use Haskell for, you may not need to define new classes often, but you will certainly be using type classes all the time. Many of the most powerful features and sophisticated capabilities of Haskell rely on type classes (residing either in the standard libraries or elsewhere). From this point on, classes will be a prominent presence in our studies.
- Data structures provide the most intuitive examples; however, there are functors which cannot reasonably be seen as data structures. A commonplace metaphor consists in thinking of functors as containers; like all metaphors, however, it can be stretched only so far.
- The functor laws, and indeed the concept of a functor, are grounded on a branch of Mathematics called Category Theory which should not be a concern for you at this point. We will have opportunities to explore related topics in the Advanced Track of this book.
- This is analogous to the gain in clarity provided by replacing explicit recursive algorithms on lists with implementations based on higher-order functions.
- Note for the curious: For one example, have a peek at Applicative Functors in the Advanced Track (for the moment you can ignore the references to monads there).