I have a question regarding the connection between monads and comonads. I always thought of the comonad operations as being the inverse operations of the corresponding monad functions.
That's not true.
If this is the case I would like to know what the corresponding monad for the following comond could be. This comonad treats lists as pointed sets where the first element is focused.
instance Comonad [] where extract = head duplicate xs = init (zipWith (++) (tails xs) (inits xs))
First of all, this is NOT a comonad. The term "comonad" doesn't refer just to instances of Comonad class, it refers to instances that satisfy several laws. Secondly, "[]" is a monad, and there is an instance of Monad class for it. It's working; don't fix it.