comonad

While musing what more useful standard class to suggest to this one class Coordinate c where createCoordinate :: x -> y -> c x y getFirst :: c x y -> x getSecond :: c x y -> y addCoordinates :: (Num x, Num y) => c x y -> c x y -> c x y it occured me that instead of something VectorSpace -y or R2 , a rather more general beast might lurk here: a Type -> Type -> Type whose two contained types can both be extracted. Hm, perhaps they can be extract ed ? Turns out neither the comonad nor bifunctors package contains something called Bicomonad . Question is, would such a class even make sense,

Can we solve this equation for X ? Applicative is to monad what X is to comonad After giving it some thought, I think this is actually a backward question. One might think that ComonadApply is to Comonad what Applicative is to Monad , but that is not the case. But to see this, let us use PureScript's typeclass hierarchy: class Functor f where fmap :: (a -> b) -> f a -> f b class Functor f => Apply f where apply :: f (a -> b) -> f a -> f b -- (<*>) class Apply f => Applicative f where pure :: a -> f a class Applicative m => Monad m where bind :: m a -> (a -> m b) -> m b -- (>>=) -- join :: m (m

Okay, so let's say you have the type newtype Dual f a = Dual {dual :: forall r. f(a -> r)->r} As it turns out, when f is a Comonad, Dual f is a Monad (fun exercise). Does it work the other way around? You can define fmap ab (Dual da) = Dual $ \fb -> da $ fmap (. ab) fb and extract (Dual da) = da $ return id , but I do not know how to define duplicate or extend . Is this even possible? If not, what the proof there is not (is there a particular Monad m for which you can prove Dual m is not a comonad)? Some observations: Dual IO a is essentially Void (and Const Void is a valid Comonad ). Dual m a

Having some idea of what the Comonad typeclass is in Haskell , I've heard about the Store comonad. But looking at Control.Comonad.Store.Lazy , I don't really get it. What does it mean? What is it for? I've heard that Store = CoState, the dual of the State Monad. What does that mean? ehird It's much easier if you look at the definition of StoreT itself . You can think of it as a "place" in a larger structure. For instance, a lens is just a -> Store b a ; you get the value of the b field, and a function b -> a to put a new value back into the larger context. Considering it in its simplified, non