Generalising ($) like Control.Category generalises (.)

Question

I had a thought to generalise ($) like Control.Category generalises (.), and I\'ve done so with the code at the end of this post (also

Answer 1

There are two abstractions used for things like this in Haskell, one usings Arrows and the other Applicatives. Both can be broken down into smaller parts than those used in base.

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If you go in the Arrow direction and break down the capabilities of Arrows into component pieces, you'd have a separate class for those arrows that are able to lift arbitrary functions into the arrow.

class ArrowArr a where
    arr :: (b -> c) -> a b c

This would be the opposite of ArrowArr, arrows where any arbitrary arrow can be dropped to a function.

class ArrowFun a where
    ($) :: a b c -> (b -> c)

If you just split arr off of Arrow you are left with arrow like categories that can construct and deconstruct tuples.

class Category a => ArrowLike a where
    fst   :: a (b, d) b
    snd   :: a (d, b) b
    (&&&) :: a b c -> a b c' -> a b (c,c')

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If you go in the Applicative direction this is a Copointed "Applicative without pure" (which goes by the name Apply).

class Copointed p where Source
    copoint :: p a -> a

class Functor f => Apply f where
  (<.>) :: f (a -> b) -> f a -> f b

When you go this way you typically drop the Category for functions and instead have a type constructor C a representing values (including function values) constructed according to a certain set of rules.