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A functor with function application within the functor

Description

Applicative functors are described here briefly in Haskell's sense.

The corresponding type class (modulo some simplifications) looks as follows.

class Functor f => Applicative f where
  pure :: a -> f a
  (<*>) :: f (a -> b) -> f a -> f b

The expectation is that pure promotes a value to a functorial value whereas "<*>" can be seen as a variation of fmap such that a function within the functor (as opposed to just a plain function) is applied to a functorial value.

The following laws are assumed.

pure f <*> x = fmap f x
pure id <*> v = v
pure (.) <*> u <*> v <*> w = u <*> (v <*> w)
pure f <*> pure x = pure (f x)
u <*> pure y = pure ($ y) <*> u

Illustration

We make Maybe and lists applicative functors:

instance Applicative Maybe where
  pure = Just
  Nothing <*> _ = Nothing
  (Just f) <*> x = fmap f x

instance Applicative [] where
  pure x = [x]
  fs <*> xs = [ f x | f <- fs, x <- xs ]

Thus, in the Maybe case, a Nothing as a function makes us return a Nothing as result, but if the function is available then it is fmapped over the argument. In the list case, we use a list comprehension to apply all available functions too all available values.

The instances can be exercised at the Haskell prompt as follows:

> Just odd <*> Just 2
Just False
> [odd, even] <*> [1,2,3,4]
[True,False,True,False,False,True,False,True]

To see that applicative functors facilitate function application for functorial values pretty well, consider the following functorial variation on plain function application.

(<$>) :: Functor f => (a -> b) -> f a -> f b
f <$> x = fmap f x

Consider the following application.

> (+) <$> [1,2] <*> [3,4]
[4,5,5,6]

Thus, the applicative operator "<*>" is used to line up (any number of) functorial arguments and fmap is used for the "rest" of the application.

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