This chapter serves a dual role. In it, we will not only introduce the very important
Functor class but also use it as a simple example of how type classes can be useful tools for solving problems in a more general way.
In Other data structures, we have seen how apply-to-all-elements operations (of which
map is the prime example) are not specific to lists. One of the examples we worked through was that of the following tree datatype:
data Tree a = Leaf a | Branch (Tree a) (Tree a) deriving (Show)
The map function we wrote for it 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)
Later in that chapter we also defined "maps" for
Weird and (in a passage of the final example)
Maybe; and could conceivably do the same for any arbitrary data structure.
map makes the apply-to-all-elements strategy general with regards to the function being applied. Now, we can see that a further generalization is possible: mapping over types other than lists. Type classes provide a way of expressing this generality.
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
fmap, it takes
a as a type parameter in one of its appearances and
b in the other. It becomes easier to see what is going on by considering an instance of
Functor - say,
Maybe. 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; and thus we didn't really need to have implemented
maybeMap for that example in "Other data structures".)
Functor instance for lists (also in Prelude) is just...
instance Functor  where fmap = map
... and replacing
 in the
fmap signature gives us the familiar type of
Summing it all up, we can say that
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)
To close this first stretch, 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). Indeed, a type whose
Functor instance does not obey the functor laws cannot properly be called a functor. 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:
- Starting from the more superficial one, 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
fmapwe instantly have a general idea of what is going on.
- More significant, however, is the ability, granted by the type class system, to 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 hierarchical libraries inherit from
Type classes make it possible to create general solutions to whole categories of problems. While it is possible that, depending on what you will use Haskell for, you will not need to define new classes often, it is sure that you will 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. And of course, classes will be a permanent presence in this book from this point on - beginning with the all-important
- Such generalization was one of the main themes of More about lists.
- 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. That need not be a concern for you at this point; in any case, we will have opportunities to explore related topics in the Advanced Track of this book.
- The situation 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).