[Haskell-cafe] Re: Monads that are Comonads and the role of Adjunction

[apfelmus]

Dan Weston wrote:
> newtype O f g a = O (f (g a))   -- Functor composition:  f `O` g
>
> instance (Functor f, Functor g) => Functor (O f g) where ...
> instance Adjunction f g         => Monad   (O g f) where ...
> instance Adjunction f g         => Comonad (O f g) where ... 
>
> class (Functor f, Functor g) => Adjunction f g | f -> g, g -> f where
>   leftAdjunct  :: (f a -> b) -> a -> g b
>   rightAdjunct :: (a -> g b) -> f a -> b
> ----------------------------------------------------------
> 
> Functors are associative but not generally commutative. Apparently a 
> Monad is also a Comonad if there exist left (f) and right (g) adjuncts 
> that commute.

Yes, but that's only sufficient, not necessary.

Jules and David already pointed out that while every monad comes from an 
adjunction, this adjunction usually involves categories different from 
Hask. So, there are probably no adjoint functors f and g in Hask such that

   [] ~= g `O` f

or

>> data L a = One a | Cons a (L a)   -- non-empty list

   L ~= g `O` f

(proof?) Yet, both are monads and the latter is even a comonad.

Moreover, f and g can only commute if they have the same source and 
target category (Hask in our case). And even when they don't commute, 
the resulting monad could still be a comonad, too.

> My category theory study stopped somewhere between Functor and 
> Adjunction, but is there any deep magic you can describe here in a 
> paragraph or two? I feel like I will never get Monad and Comonad until I 
> understand Adjunction.

Alas, I wish I could, but I have virtually no clue about adjoint 
functors myself :)

I only know the classic example that conjunction and implication

   f a = (a,S)
   g b = S -> b

are adjoint

   (a,S) -> b  =  a -> (S -> b)

which is just well-known currying. We get the state monad

   (g `O` f) a = S -> (S,a)

and the stream comonad

   (f `O` f) a = (S, S -> a)

out of that.


Regards
apfelmus