[Haskell] Typing in haskell and mathematics

[Jacques Carette]

Tomasz Zielonka <tomasz.zielonka at gmail.com> wrote:
> It's not as bad as you think. You can do this:
> 
>     {-# OPTIONS -fglasgow-exts #-}
> 
>     module Apply where
> 
>     class Apply f a b | f -> a, f -> b where
>         apply :: f -> a -> b
> 
>     instance Apply (a -> b) a b where
>         apply f a = f a
> 
>     instance Apply (a1 -> b1, a2 -> b2) (a1, a2) (b1, b2) where
>         apply (f1, f2) (a1, a2) = (f1 a1, f2 a2)
[snip]

Very nice.  But in the scrap-your-boilerplate spirit, it would be nice if one could instead say

instance* Apply (T (a -> b)) a b where
     apply (T f) a = T (f a)

where instance* is an instance template, and T is a ``shape functor'' (in the sense of polynomial functors specifying 
an y of algebra/coalgebra/bialgebra/dialgebra).  Or maybe even go for analytic functors (a la Joyal).

Well, I guess it's up to me to work out the theory... [based on the work of (at least) Jay, Hinze, Jeuring, Laemmel, 
Jansson and Peyton-Jones ! ]

Jacques