On Sun, Jun 27, 2010 at 7:43 AM, Sjoerd Visscher <sjo...@w3future.com>wrote:
> Hi Max, > > This is really interesting! > > > 1. There exist total functions: > > > >> lift :: X d => d a -> D a > >> lower :: X d => D a -> d a > > > > 2. And you can write a valid instance: > > > >> instance X D > > > > With *no superclass constraints*. > > All your examples have a more specific form: > > > lift :: X d => d a -> D d a > > lower :: X d => D d a -> d a > > instance X (D d) > > This might help when looking for a matching categorical concept. With your > original signatures I was thinking of initial/terminal objects, but that's > not the case. > > > 2. Is there a mother of all idioms? By analogy with the previous three > > examples, I tried this: > > > >> -- (<**>) :: forall a. i a -> (forall b. i (a -> b) -> i b) > >> newtype Thingy i a = Thingy { runThingy :: forall b. i (a -> b) -> i b } > > > > But I can't see how to write either pure or <*> with that data type. > > This version seems to work slightly better: > > > >> newtype Thingy i a = Thingy { runThingy :: forall b. Yoneda i (a -> b) > -> i b } > > > > Because you can write pure (pure x = Thingy (\k -> lowerYoneda (fmap > > ($ x) k))). But <*> still eludes me! > > It's usually easier to switch to Monoidal functors when playing with > Applicative. (See the original Functional Pearl "Applicative programming > with effects".) > > Then I got this: > > newtype Thingy i a = Thingy { runThingy :: forall b. Yoneda i b -> Yoneda i > (a, b) } > > (&&&) :: Thingy i c -> Thingy i d -> Thingy i (c, d) > mf &&& mx = Thingy $ fmap (\(d, (c, b)) -> ((c, d), b)) . runThingy mx . > runThingy mf > > instance Functor (Thingy i) where > fmap f m = Thingy $ fmap (first f) . runThingy m > > instance Applicative (Thingy i) where > pure x = Thingy $ fmap (x,) > mf <*> mx = fmap (\(f, x) -> f x) (mf &&& mx) > > Note that Yoneda is only there to make it possible to use fmap without the > Functor f constraint. So I'm not sure if requiring no class constraints at > all is a good requirement. It only makes things more complicated, without > providing more insights. > > I'd say that if class X requires a superclass constraint Y, then the > instance of X (D d) is allowed to have the constraint Y d. The above code > then stays the same, only with Yoneda removed and constraints added. > This is an encoding of the fact that all Functors in Haskell are strong, and that Yoneda i is a Functor for any i :: * -> *. -Edward Kmett
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