[Haskell-cafe] Re: Monads that are Comonads and the role of
Adjunction
Derek Elkins
derek.a.elkins at gmail.com
Sun Dec 16 18:45:58 EST 2007
On Sun, 2007-12-16 at 13:49 +0100, apfelmus wrote:
> 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 :)
Learn about representability. Representability is the core of category
theory. (Though, of course, it is closely related to adjunctions.)
> 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.
There is another very closely related adjunction that is less often
mentioned.
((-)->C)^op -| (-)->C
or
a -> b -> C ~ b -> a -> C
This gives rise to the monad,
M a = (a -> C) -> C
this is also exactly the comonad it gives rise to (in the op category
which ends up being the above monad in the "normal" category).
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