[Haskell-cafe] Lifting IO actions into Applicatives

[Tom Ellis]

On Tue, Oct 01, 2013 at 12:11:23PM +0300, Roman Cheplyaka wrote:
> > Shouldn't it be an *Applicative* constraint?
> > 
> >     class Applicative t => ApplicativeIO t where
> >         liftIO :: IO a -> t a
> > 
> > and require that
> > 
> >     liftIO (pure x) = pure x
> >     liftIO (f <*> x) = liftIO f <*> liftIO x
> > 
> > Seems like ApplicativeIO makes more sense than MonadIO, which is
> > unnecessarily restrictive.  With planned Functor/Applicative/Monad shuffle,
> > the former could completely replace the latter.
> 
> Agreed, this makes perfect sense. It simply says that liftIO is an
> applicative homomorphism.

Indeed.  A related question is whether, when m and m' are monads, an
applicative homomorphism between m and m' is automatically a monad
homomorphism.  That seems important in determining if one typeclass is
actually enough.

In fact we could go crazy and define

    class Functor f => FunctorIO f where
        liftIO :: IO a -> f a

and require that

    liftIO . fmap f = fmap f . liftIO

i.e. essentially that liftIO is a natural transformation between IO and f. 
I don't know whether there are simpler sufficient conditions that allow one
to determine that such an instance is also an applicative and monad
morphism.

Tom