| Copyright | (C) 2008-2015 Edward Kmett (C) 2004 Dave Menendez |
|---|---|
| License | BSD-style (see the file LICENSE) |
| Maintainer | Edward Kmett <ekmett@gmail.com> |
| Stability | provisional |
| Portability | portable |
| Safe Haskell | Safe |
| Language | Haskell2010 |
Control.Comonad
Description
Synopsis
- class Functor w => Comonad w where
- liftW :: Comonad w => (a -> b) -> w a -> w b
- 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 a b = Cokleisli {
- runCokleisli :: w a -> b
- class Functor (f :: Type -> Type) where
- (<$>) :: Functor f => (a -> b) -> f a -> f b
- ($>) :: Functor f => f a -> b -> f b
Comonads
class Functor w => Comonad w where #
There are two ways to define a comonad:
I. Provide definitions for extract and extend
satisfying these laws:
extendextract=idextract.extendf = fextendf .extendg =extend(f .extendg)
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= fextract=>=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=idfmapextract.duplicate=idduplicate.duplicate=fmapduplicate.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:
extendf =fmapf .duplicateduplicate=extendidfmapf =extend(f .extract)
These are the default definitions of extend and duplicate and
the definition of liftW respectively.
Methods
Instances
| Comonad Identity # | |
| Comonad NonEmpty # | |
| Comonad Tree # | |
| Comonad ((,) e) # | |
| Comonad (Arg e) # | |
| Comonad (Tagged s) # | |
| Comonad w => Comonad (IdentityT w) # | |
| Comonad w => Comonad (EnvT e w) # | |
| Comonad w => Comonad (StoreT s w) # | |
| (Comonad w, Monoid m) => Comonad (TracedT m w) # | |
| Monoid m => Comonad ((->) m :: Type -> Type) # | |
| (Comonad f, Comonad g) => Comonad (Sum f g) # | |
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
Combining Comonads
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:
(.)<$>u<@>v<@>w = u<@>(v<@>w)extract(p<@>q) =extractp (extractq)duplicate(p<@>q) = (<@>)<$>duplicatep<@>duplicateq
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 =constid<$>a<@>b a<@b =const<$>a<@>b
Minimal complete definition
Nothing
Methods
(<@>) :: w (a -> b) -> w a -> w b infixl 4 #
default (<@>) :: Applicative w => w (a -> b) -> w a -> w b #
Instances
| ComonadApply Identity # | |
| ComonadApply NonEmpty # | |
| ComonadApply Tree # | |
| Semigroup m => ComonadApply ((,) m) # | |
| ComonadApply w => ComonadApply (IdentityT w) # | |
| (Semigroup e, ComonadApply w) => ComonadApply (EnvT e w) # | |
| (ComonadApply w, Semigroup s) => ComonadApply (StoreT s w) # | |
| (ComonadApply w, Monoid m) => ComonadApply (TracedT m w) # | |
| Monoid m => ComonadApply ((->) m :: Type -> Type) # | |
(<@@>) :: 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
Cokleisli Arrows
Constructors
| Cokleisli | |
Fields
| |
Instances
Functors
class Functor (f :: Type -> Type) where #
A type f is a Functor if it provides a function fmap which, given any types a and b
lets you apply any function from (a -> b) to turn an f a into an f b, preserving the
structure of f. Furthermore f needs to adhere to the following:
Note, that the second law follows from the free theorem of the type fmap and
the first law, so you need only check that the former condition holds.
Minimal complete definition
Methods
fmap :: (a -> b) -> f a -> f b #
Using ApplicativeDo: 'fmap f asdo expression
do a <- as pure (f a)
with an inferred Functor constraint.
Instances
| Functor [] | Since: base-2.1 |
| Functor Maybe | Since: base-2.1 |
| Functor IO | Since: base-2.1 |
| Functor Par1 | Since: base-4.9.0.0 |
| Functor Complex | Since: base-4.9.0.0 |
| Functor Min | Since: base-4.9.0.0 |
| Functor Max | Since: base-4.9.0.0 |
| Functor First | Since: base-4.9.0.0 |
| Functor Last | Since: base-4.9.0.0 |
| Functor Option | Since: base-4.9.0.0 |
| Functor ZipList | Since: base-2.1 |
| Functor Identity | Since: base-4.8.0.0 |
| Functor First | Since: base-4.8.0.0 |
| Functor Last | Since: base-4.8.0.0 |
| Functor Dual | Since: base-4.8.0.0 |
| Functor Sum | Since: base-4.8.0.0 |
| Functor Product | Since: base-4.8.0.0 |
| Functor Down | Since: base-4.11.0.0 |
| Functor ReadP | Since: base-2.1 |
| Functor NonEmpty | Since: base-4.9.0.0 |
| Functor IntMap | |
| Functor Tree | |
| Functor Seq | |
| Functor FingerTree | |
Defined in Data.Sequence.Internal Methods fmap :: (a -> b) -> FingerTree a -> FingerTree b # (<$) :: a -> FingerTree b -> FingerTree a # | |
| Functor Digit | |
| Functor Node | |
| Functor Elem | |
| Functor ViewL | |
| Functor ViewR | |
| Functor P | Since: base-4.8.0.0 |
Defined in Text.ParserCombinators.ReadP | |
| Functor (Either a) | Since: base-3.0 |
| Functor (V1 :: Type -> Type) | Since: base-4.9.0.0 |
| Functor (U1 :: Type -> Type) | Since: base-4.9.0.0 |
| Functor ((,) a) | Since: base-2.1 |
| Functor (Array i) | Since: base-2.1 |
| Functor (Arg a) | Since: base-4.9.0.0 |
| Monad m => Functor (WrappedMonad m) | Since: base-2.1 |
Defined in Control.Applicative Methods fmap :: (a -> b) -> WrappedMonad m a -> WrappedMonad m b # (<$) :: a -> WrappedMonad m b -> WrappedMonad m a # | |
| Arrow a => Functor (ArrowMonad a) | Since: base-4.6.0.0 |
Defined in Control.Arrow Methods fmap :: (a0 -> b) -> ArrowMonad a a0 -> ArrowMonad a b # (<$) :: a0 -> ArrowMonad a b -> ArrowMonad a a0 # | |
| Functor (Proxy :: Type -> Type) | Since: base-4.7.0.0 |
| Functor (Map k) | |
| Functor f => Functor (Rec1 f) | Since: base-4.9.0.0 |
| Functor (URec Char :: Type -> Type) | Since: base-4.9.0.0 |
| Functor (URec Double :: Type -> Type) | Since: base-4.9.0.0 |
| Functor (URec Float :: Type -> Type) | Since: base-4.9.0.0 |
| Functor (URec Int :: Type -> Type) | Since: base-4.9.0.0 |
| Functor (URec Word :: Type -> Type) | Since: base-4.9.0.0 |
| Functor (URec (Ptr ()) :: Type -> Type) | Since: base-4.9.0.0 |
| Functor ((,,) a b) | Since: base-4.14.0.0 |
| Arrow a => Functor (WrappedArrow a b) | Since: base-2.1 |
Defined in Control.Applicative Methods fmap :: (a0 -> b0) -> WrappedArrow a b a0 -> WrappedArrow a b b0 # (<$) :: a0 -> WrappedArrow a b b0 -> WrappedArrow a b a0 # | |
| Functor m => Functor (Kleisli m a) | Since: base-4.14.0.0 |
| Functor (Const m :: Type -> Type) | Since: base-2.1 |
| Functor f => Functor (Ap f) | Since: base-4.12.0.0 |
| Functor f => Functor (Alt f) | Since: base-4.8.0.0 |
| (Applicative f, Monad f) => Functor (WhenMissing f x) | Since: containers-0.5.9 |
Defined in Data.IntMap.Internal Methods fmap :: (a -> b) -> WhenMissing f x a -> WhenMissing f x b # (<$) :: a -> WhenMissing f x b -> WhenMissing f x a # | |
| Functor f => Functor (Indexing f) | |
| Functor (Tagged s) | |
| Functor f => Functor (Reverse f) | Derived instance. |
| Functor (Constant a :: Type -> Type) | |
| Functor m => Functor (ReaderT r m) | |
| Functor m => Functor (IdentityT m) | |
| Functor f => Functor (Backwards f) | Derived instance. |
| Functor w => Functor (EnvT e w) # | |
| Functor w => Functor (StoreT s w) # | |
| Functor w => Functor (TracedT m w) # | |
| Functor ((->) r :: Type -> Type) | Since: base-2.1 |
| Functor (K1 i c :: Type -> Type) | Since: base-4.9.0.0 |
| (Functor f, Functor g) => Functor (f :+: g) | Since: base-4.9.0.0 |
| (Functor f, Functor g) => Functor (f :*: g) | Since: base-4.9.0.0 |
| Functor ((,,,) a b c) | Since: base-4.14.0.0 |
| (Functor f, Functor g) => Functor (Product f g) | Since: base-4.9.0.0 |
| (Functor f, Functor g) => Functor (Sum f g) | Since: base-4.9.0.0 |
| Functor f => Functor (WhenMatched f x y) | Since: containers-0.5.9 |
Defined in Data.IntMap.Internal Methods fmap :: (a -> b) -> WhenMatched f x y a -> WhenMatched f x y b # (<$) :: a -> WhenMatched f x y b -> WhenMatched f x y a # | |
| (Applicative f, Monad f) => Functor (WhenMissing f k x) | Since: containers-0.5.9 |
Defined in Data.Map.Internal Methods fmap :: (a -> b) -> WhenMissing f k x a -> WhenMissing f k x b # (<$) :: a -> WhenMissing f k x b -> WhenMissing f k x a # | |
| Functor (Cokleisli w a) # | |
| Functor f => Functor (M1 i c f) | Since: base-4.9.0.0 |
| (Functor f, Functor g) => Functor (f :.: g) | Since: base-4.9.0.0 |
| (Functor f, Functor g) => Functor (Compose f g) | Since: base-4.9.0.0 |
| Functor f => Functor (WhenMatched f k x y) | Since: containers-0.5.9 |
Defined in Data.Map.Internal Methods fmap :: (a -> b) -> WhenMatched f k x y a -> WhenMatched f k x y b # (<$) :: a -> WhenMatched f k x y b -> WhenMatched f k x y a # | |
(<$>) :: Functor f => (a -> b) -> f a -> f b infixl 4 #
An infix synonym for fmap.
The name of this operator is an allusion to $.
Note the similarities between their types:
($) :: (a -> b) -> a -> b (<$>) :: Functor f => (a -> b) -> f a -> f b
Whereas $ is function application, <$> is function
application lifted over a Functor.
Examples
Convert from a Maybe IntMaybe
Stringshow:
>>>show <$> NothingNothing>>>show <$> Just 3Just "3"
Convert from an Either Int IntEither IntString using show:
>>>show <$> Left 17Left 17>>>show <$> Right 17Right "17"
Double each element of a list:
>>>(*2) <$> [1,2,3][2,4,6]
Apply even to the second element of a pair:
>>>even <$> (2,2)(2,True)
($>) :: Functor f => f a -> b -> f b infixl 4 #
Flipped version of <$.
Using ApplicativeDo: 'as ' can be understood as the
$> bdo expression
do as pure b
with an inferred Functor constraint.
Examples
Replace the contents of a Maybe IntString:
>>>Nothing $> "foo"Nothing>>>Just 90210 $> "foo"Just "foo"
Replace the contents of an Either Int IntString, resulting in an Either
Int String
>>>Left 8675309 $> "foo"Left 8675309>>>Right 8675309 $> "foo"Right "foo"
Replace each element of a list with a constant String:
>>>[1,2,3] $> "foo"["foo","foo","foo"]
Replace the second element of a pair with a constant String:
>>>(1,2) $> "foo"(1,"foo")
Since: base-4.7.0.0