Proposal: Add Compositor class as superclass of Arrow

[Twan van Laarhoven]

Conal Elliott wrote:
 > While we're at it, how about one or more classes in between
 > Category/Compositor and Arrow?

While looking into functional references I came up with the heirarchy:

 > class Category (~>) where
 >      id :: a ~> a
 >      (.) :: (b ~> c) -> (a ~> b) -> (a ~> c)
 >
 > class Cagegory (~>) => InvArrow (~>) where
 >      -- | invArr f g may only be used if  g . f = id = f. g
 >      invArr :: (a -> b) -> (b -> a) -> (a ~> b)
 >
 > class InvArrow (~>) => RefArrow (~>) where
 >      refArr :: (a -> b) -> (b -> a -> a) -> (a ~> b)
 >
 > class RefArrow (~>) => Arrow (~>) where
 >      arr :: (a -> b) -> (a ~> b)

InvArrow allows invertible functions to be lifted to the arrow type, 
while RefArrow lifts functional references. I am not too happy about the 
names, but I like the idea itself.


apfelmus wrote:
>   class Cartesian cat where
>      fst   :: cat (a,b) a
>      snd   :: cat (a,b) b
>      (&&&) :: cat c a -> cat c b -> cat c (a,b)
> 
> with the conditions
> 
>      fst . (f &&& g) = f
>      snd . (f &&& g) = g

That is not correct, fst . (f &&& g) also has the side effects of g.

 > get first and second and that would be everything we need for arrows
 > (besides arr ). But IIRC, (&&&) does impose an "order of side effects"
 > on the arguments, so Arrows are more general

For the InvArrow of invertible functions:

 > data Invertible a b = Invertible { fun :: a -> b, inverse :: b -> a }

The functions (***), first and second are definable, while fan out, 
(&&&) is not. What I came up with was:

 > class Category (~>) => CategoryPair (~>) where
 >     first  :: (a ~> b) -> ((a,d) ~> (b,d))
 >     second :: (a ~> b) -> ((d,a) ~> (d,b))
 >     (***)  :: (a ~> b) -> (c ~> d) -> ((a,c) ~> (b,d))
 >     swap   :: (a,b) ~> (b,a)

 > class CategoryPair (~>) => CategoryFanOut (~>) where
 >     (&&&)  :: (a ~> b) -> (a ~> c) -> (a ~> (b,c))

Exactly the same problem comes up for ArrowChoice and the fan out 
operator (|||), again the class can be split in two.

I am not sure how this should be combined with Conal's deep arrows, 
perhaps fst and snd could be added to CategoryPair as well.

Twan