Last time we introduce Functor, a Functor is a container which provide a function can help another function operating the Functor. This function has a name fmap
in Haskell. Therefore, a function take a type a
as parameter(a > b
) can be lifted by fmap
to handle M a
, if M
provided a fmap
. For example, Maybe
is a Functor
, (+1)
has the type Int > Int
, fmap (+1) (Just 10)
get a result: Just 11
.
Limitation of Functor
Oh, Functor seems so powerful, but programming is simple, life is hard! In the real world, a common situation is there has many M
have to handle. For example:
replicateMaybe :: Maybe Int > Maybe a > Maybe [a]
replicateMaybe (Just len) (Just a) = Just $ replicate n a
replicateMaybe _ Nothing = Nothing
replicateMaybe Nothing _ = Nothing
Can see that we fall back to pattern matching, line 3 and 4 exclude no input. We can make it easier by extract out this pattern:
liftMaybe2 :: (a > b > c) > Maybe a > Maybe b > Maybe c
liftMaybe2 f (Just a) (Just b) = Just $ f a b
liftMaybe2 _ _ _ = Nothing
Now liftMaybe2 repliacte a b
can work just as expected. Sounds great? How about lift a > b > c > d
to M a > M b > M c > M d
. How about make a lift to another M
, e.g. List
? liftList
? It seems like boilerplate code, right?
Now we have two problems:

liftMaybe_n
problem, how to handleliftMaybe
for alln
. 
liftM
problem, how to handlelift
for differentM
.
Indeed, let's dig into fmap
again. Every function with type a > b
become M a > M b
, therefore, a > b > c
would be M a > M (b > c)
. The key point is how to make M (b > c)
applied b
.
applyMaybe :: Maybe (a > b) > Maybe a > Maybe b
applyMaybe (Just f) (Just a) = Just $ f a
applyMaybe _ _ = Nothing
Now take a look at how magic happened:
sum :: Int > Int > Int > Int
sum a b c = a + b + c
(fmap sum $ Just 1) `applyMaybe` Just 2 `applyMaybe` Just 3
 Just 6
We solve liftMaybe_n
problem! The only problem is it only works for Maybe
, to solve the problem, it's the time of class.
Applicative can help!
class Functor f => Applicative f where
pure :: a > f a
(<*>) :: f (a > b) > f a > f b
<*>
is the general version of applyMaybe
. pure
could raise a variable into the calculation in Applicative
, we also call this minimum context.
Special helper <$>
<$>
has definition as below:
(<$>) :: Functor f => (a > b) > f a > f b
(<$>) = fmap
It just an alias of fmap
to help infix syntax:
(+) <$> Just 1 <*> Just 2
 Just 3
replicate <$> Just 3 <*> Just 'x'
 Just "xxx"
replicate <$> [1, 2, 3] <*> ['x', 'y', 'z']
 ["x", "y", "z", "xx", "yy", "zz", "xxx", "yyy", "zzz"]
Conclusion
I hope this article really help you understand why we need Applicative. Next time would Monad or monoid, thanks for your read and have a good day!
Posted on by:
林子篆
I am a programming language theory lover; good at system software like Networking, OS.
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