コモナド

モナドの圏論的双対

class Functor w => Comonad w where
  extract   :: w a -> a
  duplicate :: w a -> w (w a)
  extend    :: (w b -> a) -> w b -> w a
  
  duplicate = extend id
  extend f  = fmap f . duplicate

comonad: Comonads

入門

コモナドは基点付き空間 - Functorial Blog - Comonads as Spaces
コモナドはオブジェクト - Haskell for all: Comonads are objects
コモナドを使った抽象化の威力をライフゲームで試してみた - Qiita
トーラス上のライフゲームを、List Zipperを使用して実装してみた - Qiita
コモナドを攻略する - bitterharvest’s diary
Comonadic builders

代表的なコモナド

参考: Comonads in Haskell

Env

Readerモナドの圏論的双対

type Env e a = (e, a)

instance Comonad (Env e) where
  extract = snd
  extend f w = (fst w, f w)

Traced

Writerモナドの圏論的双対

type Traced e a = e -> a

instance Monoid e => Comonad (Traced e) where
  extract m = m mempty
  extend f m = \c -> f (\c' -> m (c <> c'))

Store

Stateモナドの圏論的双対

data Store s a = Store (s -> a) s

instance Functor (Store s) where
  fmap f (Store g s) = Store (f . g) s

instance Comonad (Store s) where
  extract (Store f s) = f s
  extend f (Store g s) = Store (f . Store g) s

LensはStoreコモナドの余代数

type Lens s a = a -> Store s a

The Comonad.Reader » The State Comonad

Stream

無限の長さを持つリストと考えられる

data Stream a = Cons a (Stream a)

instance Functor Stream where
  fmap f (Cons x xs) = Cons (f xs) (fmap f xs)

instance Comonad Stream where
  extract (Cons x _) = x
  duplicate xs@(Cons _ xs') = Cons xs (duplicate xs')
  extend f xs@(Cons _ xs') = Cons (f xs) (extend f xs')

普通のリストは空リストが存在するため extract が実装できないので Comonad のインスタンスにならないことに注意

List Zipper

data Zipper a = Zipper [a] a [a]

left, right :: Zipper a -> Zipper a
left (Zipper (l:ls) c rs)) = Zipper ls l (c:rs)
right (Zipper ls c (r:rs)) = Zipper (c:ls) r rs

instance Functor Zipper where
  fmap f (Zipper ls c rs) = Zipper (fmap f ls) (f c) (fmap f rs)

iterate1 :: (a -> a) -> a -> [a]
iterate1 f = tail . iterate f

instance Comonad Zipper where
  extract (Zipper _ c _) = c
  duplicate z = Zipper (iterate1 left z) z (iterate1 right z)
  extend f z = Z (fmap f $ iterate1 left z) (f z) (fmap f $ iterate1 right z)

コモナドはZipperの一般化

Zipper

コモナドトランスフォーマー

class ComonadTrans t where
  lower :: Comonad w => t w a -> w a

圏論のコモナド

The comonad is a comonoid in the category of endofunctors.

Comonads | Bartosz Milewski’s Programming Cafe