到此为止。
It is folklore that we can have a monoidal functor in Haskell.
例如，我可以提供以下定义：
class Functor f => Monoidal f where
coherence :: (f a, f b) -> f (a, b)
One possible derivation of Monoidal from Applciative is via the Bitraversable instance for
tuples:
instance (Functor f, Applicative f) => Monoidal f where
coherence :: Applicative f => (f a, f b) -> f (a, b)
coherence = bisequence
I can also recover the defining method of Applicative like this:
(Though I am not sure if it is correct with respect to the laws.)
ap :: Monoidal f => f (a -> b) -> f a -> f b
ap f = (fmap . uncurry) ($) . coherence . (f,)
There are other functions I may define through coherence, for example:
curryA :: Monoidal f => (f (a, b) -> f c) -> f a -> f b -> f c
curryA f = (fmap . fmap) (f . coherence) (,)
然后回来。
我们可以撤消这些操作吗？
class DeMonoidal f where
decoherence :: f (a, b) -> (f a, f b)
在这方面，一些应用函子具有明显的逆函数，例如：
instance DeMonoidal Stream where
decoherence = Stream.unzip
So, now I would like to have the same nice things I had above with Monoidal, but the other way
around:
deap :: (f b -> f a) -> f (b -> a)
deap = ?
uncurryA :: DeMonoidal f => (f a -> f b -> f c) -> f (a, b) -> f c
uncurryA = ?
I came up with one possible definition of uncurryA:
contramap :: (a -> b) -> (b -> c) -> (a -> c)
contramap f g = (g . f)
uncurryA :: DeMonoidal f => (f a -> f b -> f c) -> f (a, b) -> f c
uncurryA f = (contramap decoherence . uncurry) f
— But it does not look too similar to the definition of ap I have offered above, so I am yet suspicious.
With deap I did not have any luck at all. One reason is that function application cannot
generally be reversed.
???
How can I proceed with this line of inquiry? Is there ever a definition of deap? Or is it all wrong from the very beginning?
