day-comonid: The(?) dual of Applicative

This package provides a type class named Comonoid.

class Comonad f => Comonoid f where
    coapply :: f a -> Day f f a

The name "Comonoid" should be read in a context. A functor f being Comonoid means it's a comonoid in the category of Functors equipped with Day as its tensor product.

Comonoid can be contrasted with Applicative, which is equivalent to a type class for monoids in the said category of Functors.

class Functor f => Applicative f where
    pure  :: a -> f a
    (<*>) :: f (a -> b) -> f a -> f b

-- A hypothetical type class equivalent to Applicative
class Functor f => DayMonoid f where
    pure' :: Identity a -> f a
    default pure' :: Applicative f => Identity a -> f a
    pure' = pure . runIdentity

    ap' :: Day f f a -> f a
    default ap' :: Applicative f => Day f f a -> f a
    ap' = dap

Comonoid is related to Comonad, just like Applicative is related to Monad.

Applicative is a superclass of Monad just because any Monad f instance is sufficient to implement Applicative f in a certain way.

Similarly, Comonad is a superclass of Comonoid, just because having (extract :: f a -> a) and coapply is sufficient to make f a Comonad.

Applicative	=>	Monad
a -> f a		a -> f a
Day f f a -> f a		f (f a) -> f a

Comonoid	<=	Comonad
f a -> a		f a -> a
f a -> Day f f a		f a -> f (f a)

Both of these relations are rooted in the same fact that the following conversion is possible for any Functor f and Functor g:

dayToCompose :: (Functor f, Functor g) => Day f g a -> f (g a)
dayToCompose (Day fb fc op) = fmap (\b -> fmap (op b) fc) fb