“凋零”的 Profunctor 表示是什么？

Question

This article by Chris Penner谈“枯萎光学”；可用于从结构中过滤项目的光学元件。

本文对这些光学元件使用以下“Van Laarhoven”表示：

type Wither s t a b = forall f. Alternative f => (a -> f b) -> s -> f t

大多数（如果不是全部）Van Laarhoven 光学具有等效的 profunctor 表示。例如镜头：

type Lens s t a b = forall f. Functor f => (a -> f b) -> s -> f t 

相当于：

type Lens s t a b = forall p. Strong p => p a b -> p s t

Wither 也有 Profactor 代表吗？如果是这样，它是什么？

Answer 1

克里斯在这里；这是我对 Profunctor 光学表示的看法：

Here's the profunctor class:

import Data.Profunctor
import Data.Profunctor.Traversing
import Control.Applicative

class (Traversing p) => Withering p where
  cull :: (forall f. Alternative f => (a -> f b) -> (s -> f t)) -> p a b -> p s t

instance Alternative f => Withering (Star f) where
  cull f (Star amb) = Star (f amb)

instance Monoid m => Withering (Forget m) where
  cull f (Forget h) = Forget (getAnnihilation . f (AltConst . Just . h))
    where
      getAnnihilation (AltConst Nothing) = mempty
      getAnnihilation (AltConst (Just m)) = m

newtype AltConst a b = AltConst (Maybe a)
  deriving stock (Eq, Ord, Show, Functor)

instance Monoid a => Applicative (AltConst a) where
  pure _ = (AltConst (Just mempty))
  (AltConst Nothing) <*> _ = (AltConst Nothing)
  _ <*> (AltConst Nothing) = (AltConst Nothing)
  (AltConst (Just a)) <*> (AltConst (Just b)) = AltConst (Just (a <> b))

instance (Semigroup a) => Semigroup (AltConst a x) where
  (AltConst Nothing) <> _ = (AltConst Nothing)
  _ <> (AltConst Nothing) = (AltConst Nothing)
  (AltConst (Just a)) <> (AltConst (Just b)) = AltConst (Just (a <> b))

instance (Monoid a) => Monoid (AltConst a x) where
  mempty = (AltConst (Just mempty))

instance Monoid m => Alternative (AltConst m) where
  empty = (AltConst Nothing)
  (AltConst Nothing) <|> a = a
  a <|> (AltConst Nothing) = a
  (AltConst (Just a)) <|> (AltConst (Just b)) = (AltConst (Just (a <> b)))

如果您对出现的一些光学器件感兴趣，我已经实现了其中的一些 here：

绝对有可能有其他解释或一些更简单的表示，但目前这似乎可以解决问题。如果其他人有其他想法，我很乐意看到他们！

如果您有任何其他问题，很乐意多聊！