Proposal: Add Compositor class as superclass of Arrow

[Ashley Yakeley]

Conal Elliott wrote:
> While we're at it, how about one or more classes in between 
> Category/Compositor and Arrow?  One example is DeepArrow (see wiki page 
> [1] and haddock docs [2]).  Sample instances: functions and transformers 
> of typed type representations, UIs, code, and pairs of same.  I've 
> currently structured DeepArrow as a subclass of Arrow, but most of my 
> instances do not have arr.  Any real arrow (including arr) is a 
> DeepArrow (or almost), as the haddock docs mention.

It sounds like DeepArrow should be a superclass of Arrow. Could you give 
me a good example of a type that is DeepArrow but doesn't have arr?

> DeepArrow could benefit from suggestions (perhaps refactoring), and I 
> would appreciate such input.  In any case, I imagine there's some rich, 
> useful structure between Category & Arrow, and now would be a great time 
> to explore it before settling on a new class hierarchy.

The only class that comes to mind is a subclass of Category that has arr 
but not first. But I don't know any instances that aren't Arrows. A full 
hierarchy might look something like this:

   class Category cat where
     id  :: cat a a
     (.) :: cat b c -> cat a b -> cat a c

   class (Category arr) => PreArrow arr where
     arr :: (a -> b) -> arr a b

   class (PreArrow arr) => Arrow arr where
     arrAp :: arr a (b -> c) -> arr a b -> arr a c
     (&&&) :: arr a b -> arr a c -> arr a (b,c)

   class (PreArrow arr) => CoArrow arr where
     (|||) :: arr a c -> arr b c -> arr (Either a b) c

   class (Arrow arr,CoArrow arr) => ArrowChoice arr
   instance (Arrow arr,CoArrow arr) => ArrowChoice arr

But I don't know of any useful types that are PreArrow but not Arrow, or 
CoArrow but not ArrowChoice.

-- 
Ashley Yakeley