[Haskell-cafe] Expressing "self-composable" functions at the type level

[Ignas Vyšniauskas]

Hi,

I'm wondering if there's some way to express that a function can be
composed with itself at the type level. Due to polymorphism this is not
quite the same as saying that the domain and co-domain match.

Example:

swap :: (a, b) -> (b, a)
swap . swap :: (a, b) -> (a, b)

I'd like to write a function such as (in pseudocode):

involute :: ((a -> b) &&& (b -> a)) -> (a -> a)
involute f = f . f

But I obviously can't write down the type for (a -> b) &&& (b -> a)

Perhaps this can be done with some typeclass hackery?

--
Ignas