Ok, now I see a difference, why Kleisli can be used to relate typeclasses (like Monad and ArrowApply) and Cokleisli can not:
"Kleisli m () _" = "() -> m _" is isomorphic to "m _" whereas "Cokleisli m () _" = "m _ -> ()" is not. Can somebody point out the relevant category theoretical concepts, that are at work here? On Tue, May 28, 2013 at 5:43 PM, Tom Ellis <tom-lists-haskell-cafe-2...@jaguarpaw.co.uk> wrote: > On Tue, May 28, 2013 at 05:21:58PM +0200, Johannes Gerer wrote: >> That makes sense. But why does >> >> instance Monad m => ArrowApply (Kleisli m) >> >> show that a Monad can do anything an ArrowApply can (and the two are >> thus equivalent)? > > I've tried to chase around the equivalence between these two before, and > I didn't find the algebra simple. I'll give an outline. > > In non-Haskell notation > > 1) instance Monad m => ArrowApply (Kleisli m) > > means that if "m" is a Monad then "_ -> m _" is an ArrowApply. > > 2) instance ArrowApply a => Monad (a anyType) > > means that if "_ ~> _" is an ArrowApply then "a ~> _" is a Monad. > > One direction seems easy: for a Monad m, 1) gives that "_ -> m _" is an > ArrowApply. By 2), "() -> m _" is a Monad. It is equivalent > to the Monad m we started with. > > Given an ArrowApply "_ ~> _", 2) shows that "() ~> _" is a Monad. Thus by > 1) "_ -> (() ~> _)" is an ArrowApply. I believe this should be the same > type as "_ ~> _" but I don't see how to demonstrate the isomorphsim here. > > Tom > > _______________________________________________ > Haskell-Cafe mailing list > Haskell-Cafe@haskell.org > http://www.haskell.org/mailman/listinfo/haskell-cafe _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
