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
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