[Haskell-cafe] The mother of all functors/monads/categories

[Sjoerd Visscher]

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