Accumulating errors with EitherT

Question

I have the following little mini-sample application of a web API that takes a huge JSON document and is supposed to parse it in pieces and report error messages for each of

Answer 1

We could actually code this as an arrow (Kleisli transformer).

newtype EitherAT x m a b = EitherAT { runEitherAT :: a -> m (Either x b) }

instance Monad m => Category EitherAT x m where
  id = EitherAT $ return . Right
  EitherAT a . EitherAT b
       = EitherAT $ \x -> do
              ax <- a x
              case ax of Right y -> b y
                         Left e  -> return $ Left e

instance (Monad m, Semigroup x) => Arrow EitherAT x m where
  arr f = EitherAT $ return . Right . f
  EitherAT a *** EitherAT b = EitherAT $ \(x,y) -> do
      ax <- a x
      by <- b y
      return $ case (ax,by) of
        (Right x',Right y') -> Right (x',y')
        (Left e  , Left f ) -> Left $ e <> f
        (Left e  , _      ) -> Left e
        (  _     , Left f ) ->        Left f
  first = (***id)

Only, that would violate the arrow laws (you can't rewrite a *** b to first a >>> second b without losing a's error information). But if you basically see all the Lefts as merely a debugging device, you might argue it's okay.