[Haskell-cafe] Why Kleisli composition is not in the Monad signature?

[Brent Yorgey]

On Thu, Nov 29, 2012 at 03:52:58AM +0100, Ben Franksen wrote:
> Tony Morris wrote:
> > As a side note, I think a direct superclass of Functor for Monad is not
> > a good idea, just sayin'
> > 
> > class Functor f where
> >   fmap :: (a -> b) -> f a -> f b
> > 
> > class Functor f => Apply f where
> >   (<*>) :: f (a -> b) -> f a -> f b
> > 
> > class Apply f => Bind f where
> >   (=<<) :: (a -> f b) -> f a -> f b
> > 
> > class Apply f => Applicative f where
> >   unit :: a -> f a
> > 
> > class (Applicative f, Bind f) => Monad f where
> > 
> > Same goes for Comonad (e.g. [] has (=<<) but not counit)
> > ... and again for Monoid, Category, I could go on...
> 
> Hi Tony
> 
> even though I dismissed your mentioning this on the Haskell' list, I do have 
> to admit that the proposal has a certain elegance. However, before I buy 
> into this scheme, I'd like to see some striking examples for types with 
> natural (or at least useful) Apply and Bind instances that cannot be made 
> Applicative resp. Monad. 

Try writing an Applicative instances for (Data.Map.Map k).  It can't
be done, but the Apply instance is (I would argue) both natural and useful.

> Also, it is not clear to me what laws should hold 
> for them.

http://hackage.haskell.org/package/semigroupoids defines all of these
and specifies laws, presumably derived in a principled way.

-Brent