Applicative Functor on Lists

Question

I understand that following will be:

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

I also understand that fmap is implemented as map . But how could I mentally map this computation in my head? The first time I saw this I assumed the following:

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

The second time around I though about: [[3,4,5],[2,3,4]] . But on the other hand <*> returns an fb so I knew it wouldn't be possible. So my question is, what are the mental steps to make sense out of this?

Answer 1

fs <*> xs is more-or-less [fx | f <- fs, x <- xs] [fx | f <- fs, x <- xs] . Monads have the Applicative instance

fM <*> xM = do 
  f <- fM
  x <- xM
  return (f x)

which corresponds to the list comprehension pretty directly.

Answer 2

要记住这一点，你可能会发现简单地想象<*>是× （笛卡尔积）更容易：

[a, b, c] × [x, y, z] == [a x, a y, a z, b x, b y, ...]

Answer 3

Interesting. Does an applicative always have to be the cartesian product? Or is the strategy really a matter of choice. For instance could a "zip" of two lists also be valid? ie:

[F1,F2,F3] <*> [a,b,c] == [F1 a, F2 b, F3 c]