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. greetings, -- Sjoerd Visscher http://w3future.com