I propose to refactor the Arrow class, so that GHC's arrow notation can
be a bit more general. [...]
In detail, the proposal is to change the Arrow class from [...] by introducing a new class
class PreArrow a where
premap :: (b -> b') -> a b' c -> a b c
There is a slightly more principled notion that offers up this operation, paired with an fmap-like operation on the second argument that is satisfied by all arrows -- a profunctor.
which you can find implemented here:
with misc. extras here:
Dan Piponi wrote on the connection between arrows and profunctors at the end of:
If we are going to introduce another superclass, I would much rather introduce that one, as it has a better theoretical motivation and the additional laws regarding dinatural transformations and the extra structure that follows.
For instance, profunctor composition is both well defined and useful (as are profunctor collages and traces, and there is also a useful notion of representability for profunctors).
-Edward Kmett