Proposal: Add Compositor class as superclass of Arrow

[Twan van Laarhoven]

Bertram Felgenhauer wrote:

> This is a rather finely grained hierarchy, but it's still too coarse
> to define a category of invertible functions. That category has pairs:
> it supports  first,  second  and  (***), but not  fst  and  snd.

You are right, That would suggest

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

   class CategoryPair cat => CategorySelect cat where
        fst    :: cat (a,b) a
        snd    :: cat (a,b) b

And dually

   class Category cat => CategoryChoice cat where
        mirror :: cat (Either a b) (Either b a)
        left   :: cat a b -> cat (Either a c) (Either b c)
        right  :: cat a b -> cat (Either c a) (Either c b)
        (+++)  :: cat a b -> cat c d -> cat (Either a c) (Either b d)

   class CategoryChoice cat => CategoryInject cat where
        inl    :: cat a (Either a b)
        inr    :: cat b (Either a b)

Twan