The “reader” monad

Question

OK, so the writer monad allows you to write stuff to [usually] some kind of container, and get that container back at the end. In most implementations, the \"container\" can act

Answer 1

I'm not entirely sure of what the dual of a monoid should be, but thinking of dual (probably incorrectly) as the opposite of something (simply on the basis that a Comonad is the dual of a Monad, and has all the same operations but the opposite way round). Rather than basing it on mappend and mempty I would base it on:

fold :: (Foldable f, Monoid m) => f m -> m

If we specialise f to a list here, we get:

fold :: Monoid m => [m] -> m

This seems to me to contain all of the monoid class, in particular.

mempty == fold []
mappend x y == fold [x, y]

So, then I guess the dual of this different monoid class would be:

unfold :: (Comonoid m) => m -> [m]

This is a lot like the monoid factorial class that I have seen on hackage here.

So on this basis, I think the 'reader' monad you describe would be a supply monad. The supply monad is effectively a state transformer of a list of values, so that at any point we can choose to be supplied with an item from the list. In this case, the list would be the result of unfold.supply monad

I should stress, I am no Haskell expert, nor an expert theoretician. But this is what your description made me think of.