[SOLVED] What do parentheses at the kind level mean?

What do parentheses at the kind level mean?

I understand the following kinds:

String :: *
[] :: * -> *
(->) :: * -> * -> *
(,) :: * -> * -> *

But what does this kind mean and what type might it represent?

? :: (* -> *) -> *

Solution

? :: (* -> *) -> * means that if you give to ? something of kind * -> *, you'll get back a type (something of kind *). Let's take a concrete example:

newtype IntContainer f = IC { getContainer :: f Int }

This is meant to contain Ints. I can make IntContainers with lists, or sets, or anything I want (of kind * -> *) as the underlying data structure. The thing is that f isn't a type here - it needs to have another type applied to it before it is a type. So: IntContainer needs to have something applied to it, which in turn needs a type applied to it.

ghci> :kind IntContainer
IntContainer :: (* -> *) -> *

I can make concrete types by applying things of kind * -> * to IntContainer:

ghci> ic1 = IC [1,2,3]
ic1 :: IntContainer []                           -- [] :: * -> *
ghci> ic2 = IC (Data.Set.fromList [1,2,3])
ic2 :: IntContainer Set                          -- Set :: * -> *
ghci> ic3 = IC (Data.Sequence.fromList [1,2,3])
ic3 :: IntContainer Seq                          -- Seq :: * -> *