On Sun, Jan 27, 2013 at 12:20:25AM +0000, Roman Cheplyaka wrote:
> Very nice! This can be generalized to arbitrary arrows:
>
> {-# LANGUAGE ExistentialQuantification #-}
>
> import Prelude hiding (id)
> import Control.Arrow
> import Control.Applicative
> import Control.Category
>
> data F from to b c = forall d . F (from b d) (to d c)
>
> instance (Arrow from, Arrow to) => Functor (F from to b) where
> fmap f x = pure f <*> x
>
> instance (Arrow from, Arrow to) => Applicative (F from to b) where
> pure x = F (arr $ const x) id
> F from1 to1 <*> F from2 to2 =
> F (from1 &&& from2) (to1 *** to2 >>> arr (uncurry id))
You only require that from b is Applicative, so that in turn can be
generalized:
data F g to c = forall d . F (g d) (to d c)
instance (Applicative g, Arrow to) => Functor (F g to) where
fmap f x = pure f <*> x
instance (Applicative g, Arrow to) => Applicative (F g to) where
pure x = F (pure x) id
F from1 to1 <*> F from2 to2 =
F ((,) <$> from1 <*> from2) (to1 *** to2 >>> arr (uncurry id))
> I wonder what's a categorical interpretation of F itself.
It's a variety of left Kan extension (cf section 5 of "Constructing
Applicative Functors" at MPC'2012).
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