[Haskell-cafe] traversal with an arrow

[Olaf Klinke]

> On Fri, Jun 17, 2022 at 05:01:55PM +0200, Olaf Klinke wrote:
> > Is there prior art to the following generalisation? 
> 
> >     traverseArrow :: Traversable t => a x y -> a (t x) (t y) 
> 
> Perhaps you are looking for this:
> 
>     https://github.com/tomjaguarpaw/Arrows2/issues/3#issuecomment-561973678
> 

Thanks Tom, can you explain what's behind this implementation, for
those unfamiliar with arrow syntax? I guess [*] that the Traversal type
used is a manifestation of the fact that one can transform any
traversable structure (t a) into ([a],t ()) and sort the contents of
the linked list back into their original places? This way, arbitrary
traversals can be reduced to traversals over linked lists. Then the
ArrowChoice is needed to pattern match on the list constructors? 
E. Kmett's profunctor implementation wanderA, linked to from the end of
the discussion, seems to be missing from the current
Data.Profunctor.Traversing module.

Olaf

[*] Structurally, Traversal a r b ~ Free (Const a :*: ((->) r)) b
and also their Applicative instances seem to match on first glance. In
particular, Traversal a () b ~ ([a],b) whence 
traversal :: Traversable t => t a -> Traversal a b (t b)
can be specialized to 
traversal :: Traversable t => t a -> ([a],t ())