Haskell function composition operator of type (c→d) → (a→b→c) → (a→b→d)

Question

Ordinary function composition is of the type

(.) :: (b -> c) -> (a -> b) -> a -> c

I figure this should generalize to types

Answer 1

Here's what I think is an elegant way to achieve what you want. The Functor type class gives a way to 'push' a function down into a container so you can apply it to each element using fmap. You can think of a function a -> b as a container of bs with each element indexed by an element of a. So it's natural to make this instance:

instance Functor ((->) a) where
  fmap f g = f . g

(I think you can get that by importing a suitable library but I can't remember which.)

Now the usual composition of f with g is trivially an fmap:

o1 :: (c -> d) -> (b -> c) -> (b -> d)
f `o1` g = fmap f g

A function of type a -> b -> c is a container of containers of elements of type c. So we just need to push our function f down twice. Here you go:

o2 :: (c -> d) -> (a -> (b -> c)) -> a -> (b -> d)
f `o2` g = fmap (fmap f) g

In practice you might find you don't need o1 or o2, just fmap. And if you can find the library whose location I've forgotten, you may find you can just use fmap without writ ing any additional code.