What about these two very simple type classes. Are they equivalent? (As Monad and ArrowApply)
(This actually compiles in GHC) class Pointed f where pure :: a -> f a class Unit f where unit :: f a a newtype UnitPointed f a = UnitPointed f a a instance Unit f => Pointed (UnitPointed f) where pure f = UnitPointed unit newtype Kleisli f a b = Kleisli (a -> f b) instance Pointed f => Unit (Kleisli f) where unit = Kleisli pure On Tue, May 28, 2013 at 6:05 PM, Johannes Gerer <kue...@gmail.com> wrote: > 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
