[Haskell-cafe] Functor in terms of Arrow

[Tom Schouten]

Dear HC,

Does AFunctor below have a standard name?  It's a generalization of
the Functor class in terms of Arrow instead of (->):

       fmap  :: Functor f           => (i -> o) -> f i  ->  f o
       afmap :: Arrow a, AFunctor f => a i o    -> a (f i) (f o)

It pops up in less general form (AFunctor = []) in iterated functions
(difference equations / state space models), where the arrow is the
update function parameterized by state type:

   data Iter s i o = Iter ((s,i) -> (s,o))
   instance Arrow (Iter s)

More concretely:

 > afmap :: ((s,i) -> (s, o)) -> (s,[i]) -> (s,[o])
 > afmap f (s,[]) = (s,[])
 > afmap f (s0, i:is) = (sn, o:os) where
 >  (s1, o)   = f (s0,i)
 >  (sn, os)  = afmap f (s1, is)

 > f (s,i) = (s', s') where s' = s + i
 > is = [1,1,1,1,1]
 > os = afmap f (0,is)  -- (5,[1,2,3,4,5])


Cheers,
Tom