Safe Haskell	Safe
Language	Haskell2010

Comonad

Contents

Comonad
ComonadApply
Newtypes

Synopsis

class Functor w => Comonad (w :: * -> *) where
wfix :: Comonad w => w (w a -> a) -> a
cfix :: Comonad w => (w a -> a) -> w a
kfix :: ComonadApply w => w (w a -> a) -> w a
(=>=) :: Comonad w => (w a -> b) -> (w b -> c) -> w a -> c
(=<=) :: Comonad w => (w b -> c) -> (w a -> b) -> w a -> c
(<<=) :: Comonad w => (w a -> b) -> w a -> w b
(=>>) :: Comonad w => w a -> (w a -> b) -> w b
class Comonad w => ComonadApply (w :: * -> *) where
(<@@>) :: ComonadApply w => w a -> w (a -> b) -> w b
liftW2 :: ComonadApply w => (a -> b -> c) -> w a -> w b -> w c
liftW3 :: ComonadApply w => (a -> b -> c -> d) -> w a -> w b -> w c -> w d
newtype Cokleisli (w :: k -> *) (a :: k) b :: forall k. (k -> *) -> k -> * -> * = Cokleisli {
- runCokleisli :: w a -> b
}

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
Instance details 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
Instance details 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
Instance details 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
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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 :: * -> *)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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
Instance details 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
Instance details 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
Instance details 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
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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 :: * -> *)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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 :: * -> * -> *)
Instance details 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 :: * -> * -> *)
Instance details Defined in Data.Semigroupoid Methods o :: Cokleisli w j k1 -> Cokleisli w i j -> Cokleisli w i k1 #
Comonad w => Arrow (Cokleisli w)
Instance details 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)
Instance details 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)
Instance details Defined in Control.Comonad Methods app :: Cokleisli w (Cokleisli w b c, b) c #
ComonadApply w => ArrowLoop (Cokleisli w)
Instance details Defined in Control.Comonad Methods loop :: Cokleisli w (b, d) (c, d) -> Cokleisli w b c #
Functor w => Profunctor (Cokleisli w)
Instance details 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)
Instance details 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`
Instance details 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)
Instance details Defined in Data.Profunctor.Closed Methods closed :: Cokleisli f a b -> Cokleisli f (x -> a) (x -> b) #
Functor f => Costrong (Cokleisli f)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details 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)
Instance details Defined in Data.Pointed Methods point :: a0 -> Cokleisli w a a0 #
type Corep (Cokleisli w)
Instance details Defined in Data.Profunctor.Rep type Corep (Cokleisli w) = w