Is the implementation of `<*>` based on `fmap`

Question

In Maybe applicative, <*> can be implemented based on fmap. Is it incidental, or can it be generalized to other applicative(s)?

(<*>)   ::  Maybe   (a  ->  b)  ->  Maybe   a   ->  Maybe   b
Nothing <*> _   =   Nothing
(Just   g)  <*> mx  =   fmap    g   mx

Thanks.

See also In applicative, how can `<*>` be represented in terms of `fmap_i, i=0,1,2,...`?

Answer 1

It cannot be generalized. A Functor instance is unique:

instance Functor [] where
    fmap = map

but there can be multiple valid Applicative instances for the same type constructor.

-- "Canonical" instance: [f, g] <*> [x, y] == [f x, f y, g x, g y]
instance Applicative [] where
    pure x = [x]
    [] <*> _ = []
    (f:fs) <*> xs = fmap f xs ++ (fs <*> xs)

-- Zip instance: [f, g] <*> [x, y] == [f x, g y]
instance Applicative [] where
    pure x = repeat x
    (f:fs) <*> (x:xs) = f x : (fs <*> xs)
    _ <*> _ = []

In the latter, we neither want to apply any single function from the left argument to all elements of the right, nor apply all the functions on the left to any single element on the right, making fmap useless.