This chapter is undergoing a full rewrite, and so the text will likely be a bit volatile for a while  parts of it might even be shuffled around the book. The information boxes like this one scattered through the text are for the benefit of coauthors (feel free to join!). If they are still around several months from now (July 2015), please remove them. 
When covering the vital Functor
and Monad
type classes, we glossed over a third type class: Applicative
, the class for applicative functors. Like monads, applicative functors are functors with extra laws and operations; in fact, Applicative
is an intermediate class between Functor
and Monad
. Applicative
are a widely used class with a wealth of applications (pardon the pun). It enables the eponymous applicative style, a convenient way of structuring functorial computations, and also provides means to express a number of important patterns.
Functor
recapEdit
It might be a good idea to have this recap here, even though the exercises are probably the most interesting part of the section. 
We will begin with a quick review of the Functor class chapter. Functor
is characterised by the fmap
function:
class Functor f where
fmap :: (a > b) > f a > f b
If a type has an instance of Functor
, you can use fmap
to apply a functions to values in it. Another way of describing fmap
is saying that it promotes functions to act on functorial values. To ensure fmap
works sanely, any instance of Functor
must comply with the following two laws:
fmap id = id  1st functor law
fmap (g . f) = fmap g . fmap f  2nd functor law
Maybe
, for example, has a Functor
instance, and so we can easily modify the value inside it...
Prelude> fmap negate (Just 2)
Just (2)
...as long as it exists, of course.
Prelude> fmap negate Nothing
Nothing
For extra convenience, fmap
has an infix synonym, (<$>)
. It often helps readability, and also suggests how fmap
can be seen as a different kind of function application.
Prelude> negate <$> Just 2
Just (2)
Exercises 

Define instances of

Application in functorsEdit
As useful as it is, fmap
isn't much help if we want to apply a function of two arguments to functorial values. For instance, how could we sum Just 2
and Just 3
? The brute force approach would be extracting the values from the Maybe
wrapper. That, however, would mean having to do tedious checks for Nothing
. Even worse: in a different Functor
extracting the value might not even be an option (just think about IO
).
We could use fmap
to partially apply (+)
to the first argument:
Prelude> :t (+) <$> Just 2
(+) <$> Just 2 :: Num a => Maybe (a > a)
But now we are stuck: we have a function and a value both wrapped in Maybe
, and no way of applying one to the other. What we would like to have is an operator with a type akin to f (a > b) > f a > f b
to apply functions in the context of a functor. If that operator was called (<*>)
, we would be able to write:
(+) <$> Just 2 <*> Just 3
Lo and behold  if you try that in GHCi it will just work!
Prelude> (+) <$> Just 2 <*> Just 3
Just 5
The type of (<*>)
is:
Prelude> :t (<*>)
(<*>) :: Applicative f => f (a > b) > f a > f b
(<*>)
is one of the methods of Applicative
, the type class of applicative functors  functors that support function application within their contexts. Expressions such as (+) <$> Just 2 <*> Just 3
are said to be written in applicative style, which is as close as we can get to regular function application while working with a functor. If you pretend for a moment the (<$>)
, (<*>)
and Just
aren't there, our example looks just like (+) 2 3
.
The Applicative
classEdit
The definition of Applicative
is:
class (Functor f) => Applicative f where
pure :: a > f a
(<*>) :: f (a > b) > f a > f b
Beyond (<*>)
, the class has a second method, pure
, which brings arbitrary values into the functor. As an example, let's have a look at the Maybe
instance:
instance Applicative Maybe where
pure = Just
(Just f) <*> (Just x) = Just (f x)
_ <*> _ = Nothing
It doesn't do anything surprising: pure
wraps the value with Just
; (<*>)
applies the function to the value if both exists, and results in Nothing
otherwise.
Applicative functor lawsEdit
Note
For the lack of a better shorthand, in what follows we will use the jargony word morphism to refer to the values to the left of (<*>)
, which fit the type Applicative f => f (a > b)
; that is, the functionlike things inserted into an applicative functor. "Morphism" is a term which comes from category theory and which has a much wider meaning, but that needn't concern us now.
Just like Functor
, Applicative
has a set of laws which reasonable instances should follow. They are:
pure id <*> v = v  Identity
pure f <*> pure x = pure (f x)  Homomorphism
u <*> pure y = pure ($ y) <*> u  Interchange
pure (.) <*> u <*> v <*> w = u <*> (v <*> w)  Composition
Those laws are a bit of a mouthful. They become easier to understand if you think of pure
as a way to inject values into the functor in a default, featureless way, so that the result is as close as possible to the plain value. Thus:
 The identity law says that applying the
pure id
morphism does nothing, exactly like with the plainid
function.  The homomorphism law says that applying a "pure" function to a "pure" value is the same than applying the function to the value in the normal way and then using
pure
on the result. In a sense, that meanspure
preserves function application.  The interchange law says that applying a morphism to a "pure" value
pure y
is the same as applyingpure ($ y)
to the morphism. No surprises there  as we have seen in the higher order functions chapter,($ y)
is the function that suppliesy
as argument to another function.  The composition law says that if
(<*>)
is used to compose morphisms the composition is associative, like plain function composition ^{[1]}.
There is also a bonus law about the relation between fmap
and (<*>)
:
fmap f x = pure f <*> x  fmap
Applying a "pure" function with (<*>)
is equivalent to using fmap
. This law is a consequence of the other ones, so you need not bother with proving it when writing instances of Applicative
.
Exercises 


Déja vuEdit
Parts of this season might become redundant depending on the ultimate position of this chapter, and that of the ones about monads it references. 
Does pure
remind you of anything?
pure :: Applicative f => a > f a
The only difference between that and...
return :: Monad m => a > m a
... is the class constraint. pure
and return
serve the same purpose; that is, bringing values into functors. The uncanny resemblances do not stop here. Back in the chapter about State
we mentioned a function called ap
...
ap :: (Monad m) => m (a > b) > m a > m b
... which could be used to handle functions with many arguments less painful to handle in monadic code:
allTypes :: GeneratorState (Int, Float, Char, Integer, Double, Bool, Int)
allTypes = liftM (,,,,,,) getRandom
`ap` getRandom
`ap` getRandom
`ap` getRandom
`ap` getRandom
`ap` getRandom
`ap` getRandom
ap
looks a lot like (<*>)
.
Those, of course, are not coincidences. Monad
inherits from Applicative
...
Prelude> :info Monad
class Applicative m => Monad (m :: * > *) where
etc.
... because return
and (>>=)
are enough to implement pure
and (<*>)
^{[2]}.
pure = return
(<*>) = ap
ap u v = do
f < u
x < v
return (f x)
Several other monadic functions have more general applicative versions. Here are a few of them:
Monadic  Applicative  Module (where to find the applicative version) 

(>>) 
(*>) 
Prelude (GHC 7.10+); Control.Applicative 
liftM2 
liftA2 
Control.Applicative 
mapM 
traverse 
Prelude (GHC 7.10+); Data.Traversable 
sequence 
sequenceA 
Data.Traversable 
forM_ 
for_ 
Data.Foldable 
Exercises 


A section about Alternative might fit well here. On the other hand, this is already very long as it is. Also, interactions with the MonadPlus chapter should be evaluated carefully. 
ZipList
Edit
Lists are applicative functors as well. Specialised to lists, the type of (<*>)
becomes...
[a > b] > [a] > [b]
... and so (<*>)
applies a list of functions to another list. But exactly how is that done?
The standard instance of Applicative
for lists, which follows from the Monad
instance, applies every function to every element, like an explosive version of map
.
Prelude> [(2*),(5*),(9*)] <*> [1,4,7]
[2,8,14,5,20,35,9,36,63]
Interestingly, there is another reasonable way of applying a list of functions. Instead of using every combination of functions and values, we can match each function with the value in the corresponding position in the other list. A Prelude
function which can be used for that is zipWith
:
Prelude> :t zipWith
zipWith :: (a > b > c) > [a] > [b] > [c]
Prelude> zipWith ($) [(2*),(5*),(9*)] [1,4,7]
[2,20,63]
When there are two useful possible instances for a single type, the dilemma is averted by creating a newtype
which implements one of them. In this case, we have ZipList
, which lives in Control.Applicative:
newtype ZipList a = ZipList { getZipList :: [a] }
We have already seen what <*>
should be for ziplists; all that is needed is to add the newtype
wrappers:
instance Applicative ZipList where
(ZipList fs) <*> (ZipList xs) = ZipList (zipWith ($) fs xs)
pure x = undefined  TODO
As for pure
, it is tempting to use pure x= ZipList [x]
, following the standard list instance. We can't do that, however, as it violates the applicative laws. According to the identity law:
pure id <*> v = v
Substituting (<*>)
and the suggested pure
, we get:
ZipList [id] <*> ZipList xs = ZipList xs
ZipList (zipWith ($) [id] xs) = ZipList xs
Now, suppose xs
is the infinite list [1..]
:
ZipList (zipWith ($) [id] [1..]) = ZipList [1..]
ZipList [1] = ZipList [1..]
[1] = [1..]  Obviously false!
The problem is that zipWith
produces lists whose length is that of the shortest list passed as argument, and so (ZipList [id] <*>)
will cut off all elements of the other ziplist after the first. The only way to ensure zipWith ($) fs
never removes elements is making fs
infinite. The correct pure
follows from that:
instance Applicative ZipList where
(ZipList fs) <*> (ZipList xs) = ZipList (zipWith ($) fs xs)
pure x = ZipList (repeat x)
The ZipList
applicative instance offers an alternative to all the zipN and zipWithN functions in Data.List which can be extended to any number of arguments:
>>> import Control.Applicative
>>> ZipList [(2*),(5*),(9*)] <*> ZipList [1,4,7]
ZipList {getZipList = [2,20,63]}
>>> (,,) <$> ZipList [1,4,9] <*> ZipList [2,8,1] <*> ZipList [0,0,9]
ZipList {getZipList = [(1,2,0),(4,8,0),(9,1,9)]}
>>> liftA3 (,,) (ZipList [1,4,9]) (ZipList [2,8,1]) (ZipList [0,0,9])
ZipList {getZipList = [(1,2,0),(4,8,0),(9,1,9)]}
Sequencing of effectsEdit
As we have just seen, the standard Applicative
instance for lists applies every function in one list to every element of the other. That, however, does not specify (<*>)
unambiguously. To see why, try to guess what is the result of [(2*),(3*)]<*>[4,5]
without looking at the example above or the answer just below.
Prelude> [(2*),(3*)] <*> [4,5]
 ...
[8,10,12,15]
Unless you were paying very close attention or had already analysed the implementation of (<*>)
, the odds of getting it right were about even. The other possibility would be [8,12,10,15]
. The difference is that for the first (and correct) answer the result is obtained by taking the skeleton of the first list and replacing each element by all possible combinations with elements of the second list, while for the other possibility the starting point is the second list.
In more general terms, the difference between is one of sequencing of effects. Here, by effects we mean the functorial context, as opposed to the values within the functor (some examples: the skeleton of a list, actions performed in the real world in IO
, the existence of a value in Maybe
). The existence of two legal implementations of (<*>)
for lists which only differ in the sequencing of events indicates that []
is a noncommutative applicative functor. A commutative applicative functor, by contrast, leaves no margin for ambiguity in that respect. More formally, a commutative applicative functor is one for which the following holds:
liftA2 f u v = liftA2 (flip f) v u  Commutativity
Or, equivalently,
f <$> u <*> v = flip f <$> v <*> u
By the way, if you hear about commutative monads in Haskell, the concept involved is the same, only specialised to Monad
.
Commutativity (or the lack thereof) affects other functions which are derived from (<*>)
as well. (*>)
is a clear example:
(*>) :: Applicative f => f a > f b > f b
(*>)
combines effects while preserving only the values of its second argument. For monads, it is equivalent to (>>)
. Here is a demonstration of it using Maybe
, which is commutative:
Prelude> Just 2 *> Just 3
Just 3
Prelude> Just 3 *> Just 2
Just 2
Prelude> Just 2 *> Nothing
Nothing
Prelude> Nothing *> Just 2
Nothing
Swapping the arguments does not affect the effects (that is, the being and nothingness of wrapped values). For IO
, however, swapping the arguments does reorder the effects:
Prelude> (print "foo" *> pure 2) *> (print "bar" *> pure 3)
"foo"
"bar"
3
Prelude> (print "bar" *> pure 3) *> (print "foo" *> pure 2)
"bar"
"foo"
2
The convention in Haskell is to always implement (<*>)
and other applicative operators using lefttoright sequencing. Even though this convention helps reducing confusion, it also means appearances sometimes are misleading. For instance, the (<*)
function is not flip (*>)
, as it sequences effects from left to right just like (*>)
:
Prelude> (print "foo" *> pure 2) <* (print "bar" *> pure 3)
"foo"
"bar"
2
For the same reason, (<**>) :: Applicative f => f a > f (a > b) > f b
from Control.Applicative
is not flip (<*>)
. That means it provides a way of inverting the sequencing:
>>> [(2*),(3*)] <*> [4,5]
[8,10,12,15]
>>> [4,5] <**> [(2*),(3*)]
[8,12,10,15]
An alternative is the Control.Applicative.Backwards module from transformers
, which offers a newtype
for flipping the order of effects:
newtype Backwards f a = Backwards { forwards :: f a }
>>> Backwards [(2*),(3*)] <*> Backwards [4,5]
Backwards [8,12,10,15]
Exercises 


A sliding scale of powerEdit
Functor
, Applicative
, Monad
. Three closely related functor type classes; three of the most important classes in Haskell. Though we have seen many examples of Functor
and Monad
in use, and a few of Applicative
, we have not compared them head to head yet. If we ignore pure
/return
for a moment, the characteristic methods of the three classes are:
fmap :: Functor f => (a > b) > f a > f b
(<*>) :: Applicative f => f (a > b) > f a > f b
(>>=) :: Monad m => m a > (a > m b) > m b
While those look like disparate types, we can change the picture with a few aesthetic adjustments. Let's replace fmap
by its infix synonym, (<$>)
; (>>=)
by its flipped version, (=<<)
; and tidy up the signatures a bit:
(<$>) :: Functor t => (a > b) > (t a > t b)
(<*>) :: Applicative t => t (a > b) > (t a > t b)
(=<<) :: Monad t => (a > t b) > (t a > t b)
Suddenly, the similarities are striking. fmap
, (<*>)
and (=<<)
are all mapping functions over Functor
s ^{[3]}. The differences between them are in what is being mapped over in each case:
fmap
maps arbitrary functions over functors.(<*>)
mapst (a > b)
morphisms over (applicative) functors.(=<<)
mapsa > t b
functions over (monadic) functors.
The daytoday differences in uses of Functor
, Applicative
and Monad
follow from what the types of those three mapping functions allow you to do. As you move from fmap
to (<*>)
and then to (>>=)
, you gain in power, versatility and control, at the cost of guarantees about the results. We will now slide along this scale. While doing so, we will use the contrasting terms values and context to refer to plain values within a functor and to the whatever surrounds them, respectively.
The type of fmap
ensures that it is impossible to use it to change the context, no matter which function it is given. In (a > b) > t a > t b
, the (a > b)
function has nothing to do with the t
context of the t a
functorial value, and so applying it cannot affect the context. For that reason, if you do fmap f xs
on some list xs
the number of elements of the list will never change.
Prelude> fmap (2*) [2,5,6]
[4,10,12]
That can be taken as a safety guarantee or as an unfortunate restriction, depending on what you intend. In any case, (<*>)
is clearly able to change the context:
Prelude> [(2*),(3*)] <*> [2,5,6]
[4,10,12,6,15,18]
The t (a > b)
morphism carries a context of its own, which is combined with that of the t a
functorial value. (<*>)
, however, is subject to a more subtle restriction. While t (a > b)
morphisms carry context, within them there are plain (a > b)
, which are still unable to modify the context. That means the changes to the context (<*>)
performs are fully determined by the context of its arguments, and the values have no influence over the resulting context.
Prelude> (print "foo" *> pure (2*)) <*> (print "bar" *> pure 3)
"foo"
"bar"
6
Prelude> (print "foo" *> pure 2) *> (print "bar" *> pure 3)
"foo"
"bar"
3
Prelude> (print "foo" *> pure undefined) *> (print "bar" *> pure 3)
"foo"
"bar"
3
Thus with list (<*>)
you know that the length of the resulting list will be the product of the lengths of the original lists, with IO
(<*>)
you know that all real world effect will happen as long as the evaluation terminates, and so forth.
With Monad
, however, we are in a very different game. (>>=)
takes a a > t b
function, and so it is able to create context from values. That means a lot of flexibility:
Prelude> [1,2,5] >>= \x > replicate x x
[1,2,2,5,5,5,5,5]
Prelude> [0,0,0] >>= \x > replicate x x
[]
Prelude> return 3 >>= \x > print $ if x < 10 then "Too small" else "OK"
"Too small"
Prelude> return 42 >>= \x > print $ if x < 10 then "Too small" else "OK"
"OK"
Taking advantage of the extra flexibility, however, might mean having less guarantees about, for instance, whether your functions are able to unexpectedly erase parts of a data structure for pathological inputs, or whether the control flow in your application remains intelligible. In some situations there might be performance implications as well, as the complex data dependencies monadic code makes possible might prevent useful refactorings and optimisations. All in all, it is a good idea to only use as much power as needed for the task at hand. If you do need the extra capabilities of Monad
, go right ahead; however, it is often worth it to check whether Applicative
or Functor
are sufficient.
Exercises 

The next few exercises concern the following tree data structure:

The monoidal presentationEdit
Back in Understanding monads, we saw how the Monad
class can be specified using either (>=>)
or join
instead of (>>=)
. In a similar way, Applicative
also has an alternative presentation, which might be implemented through the following type class:
class Functor f => Monoidal f where
unit :: f ()
(*&*) :: f a > f b > f (a,b)
There are deep theoretical reasons behind the name "monoidal" ^{[4]}. In any case, we can informally say that it does look a lot like a monoid: unit
provides a default functorial value whose context wraps nothing of interest, and (*&*)
combines functorial values by pairing values and combining effects. The Monoidal
formulation provides a clearer view of how Applicative
manipulates functorial contexts. Naturally, unit
and (*&*)
can be used to define pure
and (<*>)
, and viceversa.
The Applicative
laws are equivalent to the following set of laws, stated in terms of Monoidal
:
fmap snd $ unit *&* v = v  Left identity
fmap fst $ u *&* unit = u  Right identity
fmap asl $ u *&* (v *&* w) = (u *&* v) *&* w  Associativity
 asl (x, (y, z)) = ((x, y), z)
The functions to the left of the ($)
are just boilerplate to convert between equivalent types, such as b
and ((), b)
. If you ignore them, the laws are a lot less opaque than in the usual Applicative
formulation. By the way, just like for Applicative
there is a bonus law, which is guaranteed to hold in Haskell:
fmap (g *** h) (u *&* v) = fmap g u *&* fmap h v  Naturality
 g *** h = \(x, y) > (g x, h y)
Exercises 


Notes
 ↑ With plain functions, we have
h . g . f = (h . g) . f = h . (g . f)
. That is why we never bother to use parentheses in the middle of(.)
chains.  ↑ And if the
Monad
instance follows the monad laws, the resultingpure
and(<*>)
will automatically follow the applicative laws.  ↑ It is not just a question of type signatures resembling each other: the similarity has theoretical ballast. One aspect of the connection is that it is no coincidence that all three type classes have identity and composition laws.
 ↑ For extra details, follow the leads from the corresponding section of the Typeclasseopedia and the blog post by Edward Z. Yang which inspired it.