Composing Monads v. Applicative Functors

Question

The Typeclassopedia\'s Monad Transformers section explains:

Unfortunately, monads do not compose as nicely as applicative functors (yet anoth

Answer 1

There are several notions by which types of kind * -> * might "compose". The more important one is you can compose them "sequentially".

newtype Compose f g x = Compose { getCompose :: f (g x) }

Here you can see that Compose has kind (* -> *) -> (* -> *) -> (* -> *) much like any good composition of functors ought to.

So the question is: are there law-abiding instances like the following?

instance (Applicative f, Applicative g) => Applicative (Compose f g)
instance (Monad f,       Monad g)       => Monad       (Compose f g)

And the short answer as to why monads don't compose as well as applicatives is that while the first instance can be written the second cannot. Let's try!

---

We can warm up with Functor

instance (Functor f,     Functor g)     => Functor     (Compose f g) where
  fmap f (Compose fgx) = Compose (fmap (fmap f) fgx)

Here we see that because we can fmap an fmap-ed f we can pass it through the layers f and g like we need to. A similar game is played with pure

instance (Applicative f, Applicative g) => Applicative (Compose f g) where
  pure a = Compose (pure (pure a))

and while (<*>) appears tricky, if you look carefully it's the exact same trick we used with both fmap and pure.

  Compose fgf <*> Compose fgx = Compose ((<*>) <$> fgf <*> fgx)

In all cases, we can push the operators we need "through" the layers of f and g exactly as we might hope.

But now let's take a look at Monad. Instead of trying to define Monad via (>>=), I'm going to instead work via join. To implement Monad we need to implement

join :: Compose f g (Compose f g x) -> Compose f g x

using

join_f :: f (f x) -> f x  -- and
join_g :: g (g x) -> g x

or, if we strip off the newtype noise, we need

join :: f (g (f (g x))) -> f (g x)

At this point it might be clear what the problem is---we only know how to join consecutive layers of fs or gs, but here we see them interwoven. What you'll find is that we need a commutativity property

class Commute f g where
  commute :: g (f x) -> f (g x)

and now we can implement

instance (Monad f, Monad g, Commute f g) => Monad (Compose f g)

with (the newtype agnostic) join defined as

join :: f (g (f (g x))) -> f (g x)
join fgfgx = fgx where
  ffggx :: f (f (g (g x)))
  ffggx = fmap commute fgfgx
  fggx :: f (g (g x))
  fggx = join_f ffggx
  fgx :: f (g x)
  fgx = fmap join_g fggx

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So what's the upshot of all this? Applicatives always Compose, but Monads Compose only when their layers Commute.

When can we commute layers? Here are some examples

instance Commute ((->) x) ((->) y) where
  commute = flip

instance Commute ((,) x) ((,) y) where
  commute (y, (x, a)) = (x, (y, a))

instance Commute ((->) x) ((,) y) where
  commute (y, xa) = \x -> (y, xa x)

-- instance Commute ((,) x) ((->) y) does not exist; try to write yourself!
--
-- OR:
-- It turns out that you need to somehow "travel back in time" to make it
-- work...
-- 
-- instance Commute ((,) x) ((->) y) where
--   commute yxa = ( ..., \y -> let (x, a) = yxa y in a )