Comonad

Comonad

class Functor w => Comonad (w :: * -> *) where

There are two ways to define a comonad:

I. Provide definitions for extract and extend satisfying these laws:

extend extract      = id
extract . extend f  = f
extend f . extend g = extend (f . extend g)

In this case, you may simply set fmap = liftW.

These laws are directly analogous to the laws for monads and perhaps can be made clearer by viewing them as laws stating that Cokleisli composition must be associative, and has extract for a unit:

f =>= extract   = f
extract =>= f   = f
(f =>= g) =>= h = f =>= (g =>= h)

II. Alternately, you may choose to provide definitions for fmap, extract, and duplicate satisfying these laws:

extract . duplicate      = id
fmap extract . duplicate = id
duplicate . duplicate    = fmap duplicate . duplicate

In this case you may not rely on the ability to define fmap in terms of liftW.

You may of course, choose to define both duplicate and extend. In that case you must also satisfy these laws:

extend f  = fmap f . duplicate
duplicate = extend id
fmap f    = extend (f . extract)

These are the default definitions of extend and duplicate and the definition of liftW respectively.

Minimal complete definition

extract, (duplicate | extend)

Methods

extract :: w a -> a

extract . fmap f = f . extract

duplicate :: w a -> w (w a)

duplicate = extend id
fmap (fmap f) . duplicate = duplicate . fmap f

extend :: (w a -> b) -> w a -> w b

extend f = fmap f . duplicate

Instances

Comonad Identity

Defined in Control.Comonad

Methods

extract :: Identity a -> a

duplicate :: Identity a -> Identity (Identity a)

extend :: (Identity a -> b) -> Identity a -> Identity b

Comonad NonEmpty

Defined in Control.Comonad

Methods

extract :: NonEmpty a -> a

duplicate :: NonEmpty a -> NonEmpty (NonEmpty a)

extend :: (NonEmpty a -> b) -> NonEmpty a -> NonEmpty b

Comonad Tree

Defined in Control.Comonad

Methods

extract :: Tree a -> a

duplicate :: Tree a -> Tree (Tree a)

extend :: (Tree a -> b) -> Tree a -> Tree b

Comonad Log

Defined in Numeric.Log

Methods

extract :: Log a -> a

duplicate :: Log a -> Log (Log a)

extend :: (Log a -> b) -> Log a -> Log b

Comonad ((,) e)

Defined in Control.Comonad

Methods

extract :: (e, a) -> a

duplicate :: (e, a) -> (e, (e, a))

extend :: ((e, a) -> b) -> (e, a) -> (e, b)

(Representable f, Monoid (Rep f)) => Comonad (Co f)

Defined in Data.Functor.Rep

Methods

extract :: Co f a -> a

duplicate :: Co f a -> Co f (Co f a)

extend :: (Co f a -> b) -> Co f a -> Co f b

Comonad (Arg e)

Defined in Control.Comonad

Methods

extract :: Arg e a -> a

duplicate :: Arg e a -> Arg e (Arg e a)

extend :: (Arg e a -> b) -> Arg e a -> Arg e b

Comonad (Fold a)

Defined in Control.Foldl

Methods

extract :: Fold a a0 -> a0

duplicate :: Fold a a0 -> Fold a (Fold a a0)

extend :: (Fold a a0 -> b) -> Fold a a0 -> Fold a b

Functor f => Comonad (Cofree f)

Defined in Control.Comonad.Cofree

Methods

extract :: Cofree f a -> a

duplicate :: Cofree f a -> Cofree f (Cofree f a)

extend :: (Cofree f a -> b) -> Cofree f a -> Cofree f b

Comonad f => Comonad (Ap f)

Defined in Control.Applicative.Free

Methods

extract :: Ap f a -> a

duplicate :: Ap f a -> Ap f (Ap f a)

extend :: (Ap f a -> b) -> Ap f a -> Ap f b

Comonad w => Comonad (Yoneda w)

Defined in Data.Functor.Yoneda

Methods

extract :: Yoneda w a -> a

duplicate :: Yoneda w a -> Yoneda w (Yoneda w a)

extend :: (Yoneda w a -> b) -> Yoneda w a -> Yoneda w b

Monoid s => Comonad (ReifiedGetter s)

Defined in Control.Lens.Reified

Methods

extract :: ReifiedGetter s a -> a

duplicate :: ReifiedGetter s a -> ReifiedGetter s (ReifiedGetter s a)

extend :: (ReifiedGetter s a -> b) -> ReifiedGetter s a -> ReifiedGetter s b

Comonad f => Comonad (MaybeApply f)

Defined in Data.Functor.Bind.Class

Methods

extract :: MaybeApply f a -> a

duplicate :: MaybeApply f a -> MaybeApply f (MaybeApply f a)

extend :: (MaybeApply f a -> b) -> MaybeApply f a -> MaybeApply f b

Comonad w => Comonad (IdentityT w)

Defined in Control.Comonad

Methods

extract :: IdentityT w a -> a

duplicate :: IdentityT w a -> IdentityT w (IdentityT w a)

extend :: (IdentityT w a -> b) -> IdentityT w a -> IdentityT w b

(Functor f, Comonad w) => Comonad (CofreeT f w)

Defined in Control.Comonad.Trans.Cofree

Methods

extract :: CofreeT f w a -> a

duplicate :: CofreeT f w a -> CofreeT f w (CofreeT f w a)

extend :: (CofreeT f w a -> b) -> CofreeT f w a -> CofreeT f w b

(Comonad f, Comonad g) => Comonad (Day f g)

Defined in Data.Functor.Day

Methods

extract :: Day f g a -> a

duplicate :: Day f g a -> Day f g (Day f g a)

extend :: (Day f g a -> b) -> Day f g a -> Day f g b

a ~ b => Comonad (Context a b)

Defined in Control.Lens.Internal.Context

Methods

extract :: Context a b a0 -> a0

duplicate :: Context a b a0 -> Context a b (Context a b a0)

extend :: (Context a b a0 -> b0) -> Context a b a0 -> Context a b b0

(Comonad f, Monoid a) => Comonad (Static f a)

Defined in Data.Semigroupoid.Static

Methods

extract :: Static f a a0 -> a0

duplicate :: Static f a a0 -> Static f a (Static f a a0)

extend :: (Static f a a0 -> b) -> Static f a a0 -> Static f a b

Comonad (Tagged s)

Defined in Control.Comonad

Methods

extract :: Tagged s a -> a

duplicate :: Tagged s a -> Tagged s (Tagged s a)

extend :: (Tagged s a -> b) -> Tagged s a -> Tagged s b

Monoid m => Comonad ((->) m :: * -> *)

Defined in Control.Comonad

Methods

extract :: (m -> a) -> a

duplicate :: (m -> a) -> m -> m -> a

extend :: ((m -> a) -> b) -> (m -> a) -> m -> b

(Comonad f, Comonad g) => Comonad (Sum f g)

Defined in Control.Comonad

Methods

extract :: Sum f g a -> a

duplicate :: Sum f g a -> Sum f g (Sum f g a)

extend :: (Sum f g a -> b) -> Sum f g a -> Sum f g b

a ~ b => Comonad (Molten i a b)

Defined in Control.Lens.Internal.Magma

Methods

extract :: Molten i a b a0 -> a0

duplicate :: Molten i a b a0 -> Molten i a b (Molten i a b a0)

extend :: (Molten i a b a0 -> b0) -> Molten i a b a0 -> Molten i a b b0

(a ~ b, Conjoined p) => Comonad (Bazaar p a b)

Defined in Control.Lens.Internal.Bazaar

Methods

extract :: Bazaar p a b a0 -> a0

duplicate :: Bazaar p a b a0 -> Bazaar p a b (Bazaar p a b a0)

extend :: (Bazaar p a b a0 -> b0) -> Bazaar p a b a0 -> Bazaar p a b b0

(a ~ b, Conjoined p) => Comonad (Bazaar1 p a b)

Defined in Control.Lens.Internal.Bazaar

Methods

extract :: Bazaar1 p a b a0 -> a0

duplicate :: Bazaar1 p a b a0 -> Bazaar1 p a b (Bazaar1 p a b a0)

extend :: (Bazaar1 p a b a0 -> b0) -> Bazaar1 p a b a0 -> Bazaar1 p a b b0

(a ~ b, Conjoined p) => Comonad (Pretext p a b)

Defined in Control.Lens.Internal.Context

Methods

extract :: Pretext p a b a0 -> a0

duplicate :: Pretext p a b a0 -> Pretext p a b (Pretext p a b a0)

extend :: (Pretext p a b a0 -> b0) -> Pretext p a b a0 -> Pretext p a b b0

(a ~ b, Conjoined p) => Comonad (BazaarT p g a b)

Defined in Control.Lens.Internal.Bazaar

Methods

extract :: BazaarT p g a b a0 -> a0

duplicate :: BazaarT p g a b a0 -> BazaarT p g a b (BazaarT p g a b a0)

extend :: (BazaarT p g a b a0 -> b0) -> BazaarT p g a b a0 -> BazaarT p g a b b0

(a ~ b, Conjoined p) => Comonad (BazaarT1 p g a b)

Defined in Control.Lens.Internal.Bazaar

Methods

extract :: BazaarT1 p g a b a0 -> a0

duplicate :: BazaarT1 p g a b a0 -> BazaarT1 p g a b (BazaarT1 p g a b a0)

extend :: (BazaarT1 p g a b a0 -> b0) -> BazaarT1 p g a b a0 -> BazaarT1 p g a b b0

(a ~ b, Conjoined p) => Comonad (PretextT p g a b)

Defined in Control.Lens.Internal.Context

Methods

extract :: PretextT p g a b a0 -> a0

duplicate :: PretextT p g a b a0 -> PretextT p g a b (PretextT p g a b a0)

extend :: (PretextT p g a b a0 -> b0) -> PretextT p g a b a0 -> PretextT p g a b b0

wfix :: Comonad w => w (w a -> a) -> a

Comonadic fixed point à la David Menendez

cfix :: Comonad w => (w a -> a) -> w a

Comonadic fixed point à la Dominic Orchard

kfix :: ComonadApply w => w (w a -> a) -> w a

Comonadic fixed point à la Kenneth Foner:

This is the evaluate function from his "Getting a Quick Fix on Comonads" talk.

(=>=) :: Comonad w => (w a -> b) -> (w b -> c) -> w a -> c infixr 1

Left-to-right Cokleisli composition

(=<=) :: Comonad w => (w b -> c) -> (w a -> b) -> w a -> c infixr 1

Right-to-left Cokleisli composition

(<<=) :: Comonad w => (w a -> b) -> w a -> w b infixr 1

extend in operator form

(=>>) :: Comonad w => w a -> (w a -> b) -> w b infixl 1

extend with the arguments swapped. Dual to >>= for a Monad.

ComonadApply

class Comonad w => ComonadApply (w :: * -> *) where

ComonadApply is to Comonad like Applicative is to Monad.

Mathematically, it is a strong lax symmetric semi-monoidal comonad on the category Hask of Haskell types. That it to say that w is a strong lax symmetric semi-monoidal functor on Hask, where both extract and duplicate are symmetric monoidal natural transformations.

Laws:

(.) <$> u <@> v <@> w = u <@> (v <@> w)
extract (p <@> q) = extract p (extract q)
duplicate (p <@> q) = (<@>) <$> duplicate p <@> duplicate q

If our type is both a ComonadApply and Applicative we further require

(<*>) = (<@>)

Finally, if you choose to define (<@) and (@>), the results of your definitions should match the following laws:

a @> b = const id <$> a <@> b
a <@ b = const <$> a <@> b

Methods

(<@>) :: w (a -> b) -> w a -> w b infixl 4

(@>) :: w a -> w b -> w b infixl 4

(<@) :: w a -> w b -> w a infixl 4

Instances

ComonadApply Identity

Defined in Control.Comonad

Methods

(<@>) :: Identity (a -> b) -> Identity a -> Identity b

(@>) :: Identity a -> Identity b -> Identity b

(<@) :: Identity a -> Identity b -> Identity a

ComonadApply NonEmpty

Defined in Control.Comonad

Methods

(<@>) :: NonEmpty (a -> b) -> NonEmpty a -> NonEmpty b

(@>) :: NonEmpty a -> NonEmpty b -> NonEmpty b

(<@) :: NonEmpty a -> NonEmpty b -> NonEmpty a

ComonadApply Tree

Defined in Control.Comonad

Methods

(<@>) :: Tree (a -> b) -> Tree a -> Tree b

(@>) :: Tree a -> Tree b -> Tree b

(<@) :: Tree a -> Tree b -> Tree a

ComonadApply Log

Defined in Numeric.Log

Methods

(<@>) :: Log (a -> b) -> Log a -> Log b

(@>) :: Log a -> Log b -> Log b

(<@) :: Log a -> Log b -> Log a

Semigroup m => ComonadApply ((,) m)

Defined in Control.Comonad

Methods

(<@>) :: (m, a -> b) -> (m, a) -> (m, b)

(@>) :: (m, a) -> (m, b) -> (m, b)

(<@) :: (m, a) -> (m, b) -> (m, a)

ComonadApply f => ComonadApply (Cofree f)

Defined in Control.Comonad.Cofree

Methods

(<@>) :: Cofree f (a -> b) -> Cofree f a -> Cofree f b

(@>) :: Cofree f a -> Cofree f b -> Cofree f b

(<@) :: Cofree f a -> Cofree f b -> Cofree f a

Monoid s => ComonadApply (ReifiedGetter s)

Defined in Control.Lens.Reified

Methods

(<@>) :: ReifiedGetter s (a -> b) -> ReifiedGetter s a -> ReifiedGetter s b

(@>) :: ReifiedGetter s a -> ReifiedGetter s b -> ReifiedGetter s b

(<@) :: ReifiedGetter s a -> ReifiedGetter s b -> ReifiedGetter s a

ComonadApply w => ComonadApply (IdentityT w)

Defined in Control.Comonad

Methods

(<@>) :: IdentityT w (a -> b) -> IdentityT w a -> IdentityT w b

(@>) :: IdentityT w a -> IdentityT w b -> IdentityT w b

(<@) :: IdentityT w a -> IdentityT w b -> IdentityT w a

(ComonadApply f, ComonadApply g) => ComonadApply (Day f g)

Defined in Data.Functor.Day

Methods

(<@>) :: Day f g (a -> b) -> Day f g a -> Day f g b

(@>) :: Day f g a -> Day f g b -> Day f g b

(<@) :: Day f g a -> Day f g b -> Day f g a

Monoid m => ComonadApply ((->) m :: * -> *)

Defined in Control.Comonad

Methods

(<@>) :: (m -> a -> b) -> (m -> a) -> m -> b

(@>) :: (m -> a) -> (m -> b) -> m -> b

(<@) :: (m -> a) -> (m -> b) -> m -> a

(a ~ b, Conjoined p) => ComonadApply (Bazaar p a b)

Defined in Control.Lens.Internal.Bazaar

Methods

(<@>) :: Bazaar p a b (a0 -> b0) -> Bazaar p a b a0 -> Bazaar p a b b0

(@>) :: Bazaar p a b a0 -> Bazaar p a b b0 -> Bazaar p a b b0

(<@) :: Bazaar p a b a0 -> Bazaar p a b b0 -> Bazaar p a b a0

(a ~ b, Conjoined p) => ComonadApply (Bazaar1 p a b)

Defined in Control.Lens.Internal.Bazaar

Methods

(<@>) :: Bazaar1 p a b (a0 -> b0) -> Bazaar1 p a b a0 -> Bazaar1 p a b b0

(@>) :: Bazaar1 p a b a0 -> Bazaar1 p a b b0 -> Bazaar1 p a b b0

(<@) :: Bazaar1 p a b a0 -> Bazaar1 p a b b0 -> Bazaar1 p a b a0

(a ~ b, Conjoined p) => ComonadApply (BazaarT p g a b)

Defined in Control.Lens.Internal.Bazaar

Methods

(<@>) :: BazaarT p g a b (a0 -> b0) -> BazaarT p g a b a0 -> BazaarT p g a b b0

(@>) :: BazaarT p g a b a0 -> BazaarT p g a b b0 -> BazaarT p g a b b0

(<@) :: BazaarT p g a b a0 -> BazaarT p g a b b0 -> BazaarT p g a b a0

(a ~ b, Conjoined p) => ComonadApply (BazaarT1 p g a b)

Defined in Control.Lens.Internal.Bazaar

Methods

(<@>) :: BazaarT1 p g a b (a0 -> b0) -> BazaarT1 p g a b a0 -> BazaarT1 p g a b b0

(@>) :: BazaarT1 p g a b a0 -> BazaarT1 p g a b b0 -> BazaarT1 p g a b b0

(<@) :: BazaarT1 p g a b a0 -> BazaarT1 p g a b b0 -> BazaarT1 p g a b a0

(<@@>) :: ComonadApply w => w a -> w (a -> b) -> w b infixl 4

A variant of <@> with the arguments reversed.

liftW2 :: ComonadApply w => (a -> b -> c) -> w a -> w b -> w c

Lift a binary function into a Comonad with zipping

liftW3 :: ComonadApply w => (a -> b -> c -> d) -> w a -> w b -> w c -> w d

Lift a ternary function into a Comonad with zipping

Newtypes

newtype Cokleisli (w :: k -> *) (a :: k) b :: forall k. (k -> *) -> k -> * -> *

The Cokleisli Arrows of a given Comonad

Constructors

Cokleisli

Fields

• runCokleisli :: w a -> b

Instances

Comonad w => Category (Cokleisli w :: * -> * -> *)

Defined in Control.Comonad

Methods

id :: Cokleisli w a a

(.) :: Cokleisli w b c -> Cokleisli w a b -> Cokleisli w a c

Extend w => Semigroupoid (Cokleisli w :: * -> * -> *)

Defined in Data.Semigroupoid

Methods

o :: Cokleisli w j k1 -> Cokleisli w i j -> Cokleisli w i k1

Comonad w => Arrow (Cokleisli w)

Defined in Control.Comonad

Methods

arr :: (b -> c) -> Cokleisli w b c

first :: Cokleisli w b c -> Cokleisli w (b, d) (c, d)

second :: Cokleisli w b c -> Cokleisli w (d, b) (d, c)

(***) :: Cokleisli w b c -> Cokleisli w b' c' -> Cokleisli w (b, b') (c, c')

(&&&) :: Cokleisli w b c -> Cokleisli w b c' -> Cokleisli w b (c, c')

Comonad w => ArrowChoice (Cokleisli w)

Defined in Control.Comonad

Methods

left :: Cokleisli w b c -> Cokleisli w (Either b d) (Either c d)

right :: Cokleisli w b c -> Cokleisli w (Either d b) (Either d c)

(+++) :: Cokleisli w b c -> Cokleisli w b' c' -> Cokleisli w (Either b b') (Either c c')

(|||) :: Cokleisli w b d -> Cokleisli w c d -> Cokleisli w (Either b c) d

Comonad w => ArrowApply (Cokleisli w)

Defined in Control.Comonad

Methods

app :: Cokleisli w (Cokleisli w b c, b) c

ComonadApply w => ArrowLoop (Cokleisli w)

Defined in Control.Comonad

Methods

loop :: Cokleisli w (b, d) (c, d) -> Cokleisli w b c

Functor w => Profunctor (Cokleisli w)

Defined in Data.Profunctor.Unsafe

Methods

dimap :: (a -> b) -> (c -> d) -> Cokleisli w b c -> Cokleisli w a d

lmap :: (a -> b) -> Cokleisli w b c -> Cokleisli w a c

rmap :: (b -> c) -> Cokleisli w a b -> Cokleisli w a c

(#.) :: Coercible c b => (b -> c) -> Cokleisli w a b -> Cokleisli w a c

(.#) :: Coercible b a => Cokleisli w b c -> (a -> b) -> Cokleisli w a c

Functor w => Corepresentable (Cokleisli w)

Defined in Data.Profunctor.Rep

Associated Types

type Corep (Cokleisli w) :: * -> *

Methods

cotabulate :: (Corep (Cokleisli w) d -> c) -> Cokleisli w d c

Comonad w => Choice (Cokleisli w)

extract approximates costrength

Defined in Data.Profunctor.Choice

Methods

left' :: Cokleisli w a b -> Cokleisli w (Either a c) (Either b c)

right' :: Cokleisli w a b -> Cokleisli w (Either c a) (Either c b)

Functor f => Closed (Cokleisli f)

Defined in Data.Profunctor.Closed

Methods

closed :: Cokleisli f a b -> Cokleisli f (x -> a) (x -> b)

Functor f => Costrong (Cokleisli f)

Defined in Data.Profunctor.Strong

Methods

unfirst :: Cokleisli f (a, d) (b, d) -> Cokleisli f a b

unsecond :: Cokleisli f (d, a) (d, b) -> Cokleisli f a b

Monad (Cokleisli w a)

Defined in Control.Comonad

Methods

(>>=) :: Cokleisli w a a0 -> (a0 -> Cokleisli w a b) -> Cokleisli w a b

(>>) :: Cokleisli w a a0 -> Cokleisli w a b -> Cokleisli w a b

return :: a0 -> Cokleisli w a a0

fail :: String -> Cokleisli w a a0

Functor (Cokleisli w a)

Defined in Control.Comonad

Methods

fmap :: (a0 -> b) -> Cokleisli w a a0 -> Cokleisli w a b

(<$) :: a0 -> Cokleisli w a b -> Cokleisli w a a0

Applicative (Cokleisli w a)

Defined in Control.Comonad

Methods

pure :: a0 -> Cokleisli w a a0

(<*>) :: Cokleisli w a (a0 -> b) -> Cokleisli w a a0 -> Cokleisli w a b

liftA2 :: (a0 -> b -> c) -> Cokleisli w a a0 -> Cokleisli w a b -> Cokleisli w a c

(*>) :: Cokleisli w a a0 -> Cokleisli w a b -> Cokleisli w a b

(<*) :: Cokleisli w a a0 -> Cokleisli w a b -> Cokleisli w a a0

Apply (Cokleisli w a)

Defined in Data.Functor.Bind.Class

Methods

(<.>) :: Cokleisli w a (a0 -> b) -> Cokleisli w a a0 -> Cokleisli w a b

(.>) :: Cokleisli w a a0 -> Cokleisli w a b -> Cokleisli w a b

(<.) :: Cokleisli w a a0 -> Cokleisli w a b -> Cokleisli w a a0

liftF2 :: (a0 -> b -> c) -> Cokleisli w a a0 -> Cokleisli w a b -> Cokleisli w a c

Pointed (Cokleisli w a)

Defined in Data.Pointed

Methods

point :: a0 -> Cokleisli w a a0

type Corep (Cokleisli w)

Defined in Data.Profunctor.Rep

type Corep (Cokleisli w) = w