Arrow Classes

[Ashley Yakeley]

In article <200306271927.36405.wolfgang@jeltsch.net>,
 Wolfgang Jeltsch <wolfgang@jeltsch.net> wrote:

> This brings me to another point. One year ago we had a discussion on The 
> Haskell Mailing List concerning arrows. (The subject of the mails was just 
> "arrows".) The point was that it seemed strange to me that first and second 
> are included in the basic arrow class Arrow while left and right have their 
> extra class ArrowChoice. Not only that it seemed strange to me but it made it 
> impossible to make Parser baseMonad an instance of Arrow. Parser baseMonad 
> has nice implementations of pure and (>>>) but none of first or second.

I agree. My own Arrow module hierarchy looks more or less like this:

  class Compositor comp where
    identity :: comp a a
    compose :: comp b c -> comp a b -> comp a c

  class (Compositor arrow) => Arrow arrow where
    arrFunction :: (p -> q) -> arrow p q

  -- | corresponds to Hughes\' \'Arrow\'
  class (Arrow arrow) => ProductArrow arrow where
    arrApply :: arrow p (q -> r) -> arrow p q -> arrow p r
    arrProduct :: arrow p q -> arrow p r -> arrow p (q,r)
    arrProduct = liftA2 (,)
 
  class (Arrow arrow) => CoproductArrow arrow where
    arrCoproduct :: arrow p r -> arrow q r -> arrow (Either p q) r

  -- | corresponds to Hughes\' \'ArrowChoice\'
  class (ProductArrow arrow,CoproductArrow arrow) => FullArrow arrow
  instance (ProductArrow arrow,CoproductArrow arrow) => FullArrow arrow

  class (Arrow arrow) => ArrowFix arrow where
    arrFix :: arrow (p,q) q -> arrow p q

  class (FullArrow arrow) => ApplyArrow arrow where
    arrApplyArrow :: arrow (arrow p q,p) q

Note the symmetry between ProductArrow and CoproductArrow.

See 
<http://cvs.sourceforge.net/cgi-bin/viewcvs.cgi/*checkout*/hbase/Source/H
Base/Category/Arrow.hs?rev=HEAD&content-type=text/plain> for all the 
details.

-- 
Ashley Yakeley, Seattle WA