Question

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?

Answer 1

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-transformer form:

data Store s a = Store (s -> a) s

instance Functor (Store s) where
  fmap f (Store g s) = Store (f . g) s

instance Extend (Store s) where
  duplicate (Store f s) = Store (Store f) s

instance Comonad (Store s) where
  extract (Store f s) = f s

i.e. duplicate changes the s -> a into an s -> Store s a that just returns the “updated” place after replacing the value, and extract restores the original a by placing the value back into the larger structure.

As far as its relation to State goes, you could look at it like this:

type State s a = s -> (a, s)
type Store s a = (s -> a, s)