On Mon, Jun 28, 2004 at 02:59:39PM +0100, Graham Klyne wrote:
Is there a function that switches monadic layering?
f :: (Monad m1,Monad m2) => m1 (m2 a) -> m2 (m1 a)
Does this even make sense in the general case? I'm thinking along the lines of a generalization of sequence to non-list monads.
Such a function is called a distributive law of m1 over m2 if it satisfies the following equations (subscripts added for readability): f . return_1 = liftM_2 return_1 f . liftM_1 return_2 = return_2 f . join_1 = liftM_2 join_1 . f . liftM_1 f f . liftM_1 join_2 = join_2 . liftM_2 f . f The composition m2.m1 can be made a monad in a standard way: return = return_2 . return_1 join = liftM_2 join_1 . join_2 . liftM_2 f if and only if such a distributive law exists. See section 9.2 of "Toposes, Triples and Theories", by Michael Barr and Charles Wells, online at http://www.cwru.edu/artsci/math/wells/pub/ttt.html ("triple" is another name for a monad)
[...], and sequence does for the List monad what I am seeking to generalize:
sequence :: [m a] -> m [a]
Yes, it's a distributive law of [] over m, but only if m is a commutative monad (lists are so sequential). Some more distributive laws: * the exception monad distributes over any other monad: either (return . Left) (liftM Right) :: Monad m => Either x (m a) -> m (Either x a) * so does the writer monad: uncurry (liftM . (,)) :: (Monoid w, Monad m) => (w, m a) -> m (w,a) * any monad distributes over the reader monad: flip (liftM . flip ($)) :: Monad m => m (r -> a) -> r -> m a