Dan Weston wrote:
apfelmus wrote:
Luke Palmer wrote:
Isn't a type which is both a Monad and a Comonad just Identity?
(I'm actually not sure, I'm just conjecting)
Good idea, but it's not the case.
data L a = One a | Cons a (L a) -- non-empty list
Maybe I can entice you to elaborate slightly. From
http://www.eyrie.org/~zednenem/2004/hsce/Control.Functor.html and
Control.Comonad.html there is
----------------------------------------------------------
newtype 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 ...
-- I assume Haskell can infer Functor (O g f) from Monad (O g f), which
-- is why that is missing here?
No. But it can infer Functor (O g f) from instance (Functor f, Functor
g) => Functor (O f g), (using 'g' for 'f' and 'f' for 'g').
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. [and only if also??? Is there a contrary example of a
Monad/Comonad for which no such f and g exist?]
In the case of
> data L a = One a | Cons a (L a) -- non-empty list
what are the appropriate definitions of leftAdjunct and rightAdjunct?
Are they Monad.return and Comonad.extract respectively? That seems to
unify a and b unnecessarily. Do they wrap bind and cobind? Are they of
any practical utility?
I think you're asking the wrong question!
The first question needs to be :
What is "f" and what is "g" ? What are the two Functors in this case?
We know that we want g `O` f to be L, because we know that the unit is
return, i.e. One, and
unit :: a -> O g f a
otherwise known as eta :: a -> O g f a
We also know there is a co-unit epsilon :: O f g a -> a, but we don't
know much about that until we work out how to decompose L into two Functors.
There are two standard ways to decompose a monad into two adjoint
functors: the Kleisli decomposition and the Eilenberg-Moore decomposition.
However, neither of these categories is a subcategory of Hask in an
obvious way, so I don't immediately see how to write "f" and "g" as
haskell functors.
Maybe someone else can show the way :)
Jules
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