| 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 Source #
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.
Instances
| Comonad Identity Source # | |
| Comonad NonEmpty Source # | |
| Comonad Tree Source # | |
| Comonad (Arg e) Source # | |
| Comonad ((,) e) Source # | |
| Comonad w => Comonad (EnvT e w) Source # | |
| Comonad w => Comonad (StoreT s w) Source # | |
| (Comonad w, Monoid m) => Comonad (TracedT m w) Source # | |
| Comonad (Tagged s) Source # | |
| Comonad w => Comonad (IdentityT w) Source # | |
| (Comonad f, Comonad g) => Comonad (Sum f g) Source # | |
| Monoid m => Comonad ((->) m) Source # | |
kfix :: ComonadApply w => w (w a -> a) -> w a Source #
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 Source #
Left-to-right Cokleisli composition
(=<=) :: Comonad w => (w b -> c) -> (w a -> b) -> w a -> c infixr 1 Source #
Right-to-left Cokleisli composition
Combining Comonads
class Comonad w => ComonadApply w where Source #
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 Source #
default (<@>) :: Applicative w => w (a -> b) -> w a -> w b Source #
Instances
| ComonadApply Identity Source # | |
| ComonadApply NonEmpty Source # | |
| ComonadApply Tree Source # | |
| Semigroup m => ComonadApply ((,) m) Source # | |
| (Semigroup e, ComonadApply w) => ComonadApply (EnvT e w) Source # | |
| (ComonadApply w, Semigroup s) => ComonadApply (StoreT s w) Source # | |
| (ComonadApply w, Monoid m) => ComonadApply (TracedT m w) Source # | |
| ComonadApply w => ComonadApply (IdentityT w) Source # | |
| Monoid m => ComonadApply ((->) m) Source # | |
(<@@>) :: ComonadApply w => w a -> w (a -> b) -> w b infixl 4 Source #
A variant of <@> with the arguments reversed.
liftW2 :: ComonadApply w => (a -> b -> c) -> w a -> w b -> w c Source #
Lift a binary function into a Comonad with zipping
liftW3 :: ComonadApply w => (a -> b -> c -> d) -> w a -> w b -> w c -> w d Source #
Lift a ternary function into a Comonad with zipping
Cokleisli Arrows
newtype Cokleisli w a b Source #
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.
See https://www.schoolofhaskell.com/user/edwardk/snippets/fmap or
https://github.com/quchen/articles/blob/master/second_functor_law.md
for an explanation.
Minimal complete definition
Methods
fmap :: (a -> b) -> f a -> f b #
fmap is used to apply a function of type (a -> b) to a value of type f a,
where f is a functor, to produce a value of type f b.
Note that for any type constructor with more than one parameter (e.g., Either),
only the last type parameter can be modified with fmap (e.g., b in `Either a b`).
Some type constructors with two parameters or more have a Bifunctor
Examples
Convert from a Maybe IntMaybe String
using show:
>>>fmap show NothingNothing>>>fmap show (Just 3)Just "3"
Convert from an Either Int IntEither Int String using show:
>>>fmap show (Left 17)Left 17>>>fmap show (Right 17)Right "17"
Double each element of a list:
>>>fmap (*2) [1,2,3][2,4,6]
Apply even to the second element of a pair:
>>>fmap even (2,2)(2,True)
It may seem surprising that the function is only applied to the last element of the tuple
compared to the list example above which applies it to every element in the list.
To understand, remember that tuples are type constructors with multiple type parameters:
a tuple of 3 elements (a,b,c) can also be written (,,) a b c and its Functor instance
is defined for Functor ((,,) a b) (i.e., only the third parameter is free to be mapped over
with fmap).
It explains why fmap can be used with tuples containing values of different types as in the
following example:
>>>fmap even ("hello", 1.0, 4)("hello",1.0,True)
Instances
| Functor ZipList | Since: base-2.1 |
| Functor Complex | Since: base-4.9.0.0 |
| 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 First | Since: base-4.9.0.0 |
| Functor Last | Since: base-4.9.0.0 |
| Functor Max | Since: base-4.9.0.0 |
| Functor Min | Since: base-4.9.0.0 |
| Functor Dual | Since: base-4.8.0.0 |
| Functor Product | Since: base-4.8.0.0 |
| Functor Sum | Since: base-4.8.0.0 |
| Functor NonEmpty | Since: base-4.9.0.0 |
| Functor Par1 | Since: base-4.9.0.0 |
| Functor P | Since: base-4.8.0.0 |
Defined in Text.ParserCombinators.ReadP | |
| Functor ReadP | Since: base-2.1 |
| Functor IntMap | |
| Functor Digit | |
| Functor Elem | |
| Functor FingerTree | |
Defined in Data.Sequence.Internal Methods fmap :: (a -> b) -> FingerTree a -> FingerTree b # (<$) :: a -> FingerTree b -> FingerTree a # | |
| Functor Node | |
| Functor Seq | |
| Functor ViewL | |
| Functor ViewR | |
| Functor Tree | |
| Functor IO | Since: base-2.1 |
| Functor AnnotDetails | |
Defined in Text.PrettyPrint.Annotated.HughesPJ Methods fmap :: (a -> b) -> AnnotDetails a -> AnnotDetails b # (<$) :: a -> AnnotDetails b -> AnnotDetails a # | |
| Functor Doc | |
| Functor Span | |
| Functor Q | |
| Functor TyVarBndr | |
| Functor Maybe | Since: base-2.1 |
| Functor Solo | Since: base-4.15 |
| Functor List | Since: base-2.1 |
| 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 (Either a) | Since: base-3.0 |
| Functor (Proxy :: Type -> Type) | Since: base-4.7.0.0 |
| Functor (Arg a) | Since: base-4.9.0.0 |
| Functor (Array i) | Since: base-2.1 |
| Functor (U1 :: Type -> Type) | Since: base-4.9.0.0 |
| Functor (V1 :: Type -> Type) | Since: base-4.9.0.0 |
| Functor (Map k) | |
| Functor ((,) a) | Since: base-2.1 |
| 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 # (<$) :: a0 -> WrappedArrow a b b0 -> WrappedArrow a b a0 # | |
| Functor m => Functor (Kleisli m a) | Since: base-4.14.0.0 |
| Functor (Const m :: Type -> Type) | Since: base-2.1 |
| Functor f => Functor (Ap f) | Since: base-4.12.0.0 |
| Functor f => Functor (Alt f) | Since: base-4.8.0.0 |
| (Generic1 f, Functor (Rep1 f)) => Functor (Generically1 f) | Since: base-4.17.0.0 |
Defined in GHC.Generics Methods fmap :: (a -> b) -> Generically1 f a -> Generically1 f b # (<$) :: a -> Generically1 f b -> Generically1 f a # | |
| Functor f => Functor (Rec1 f) | Since: base-4.9.0.0 |
| Functor (URec (Ptr ()) :: Type -> Type) | 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 w => Functor (EnvT e w) Source # | |
| Functor w => Functor (StoreT s w) Source # | |
| Functor w => Functor (TracedT m w) Source # | |
| (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 # (<$) :: a -> WhenMissing f x b -> WhenMissing f x a # | |
| Functor f => Functor (Indexing f) | |
| Functor (Tagged s) | |
| Functor f => Functor (Backwards f) | Derived instance. |
| Functor m => Functor (IdentityT m) | |
| Functor m => Functor (ReaderT r m) | |
| Functor (Constant a :: Type -> Type) | |
| Functor f => Functor (Reverse f) | Derived instance. |
| Functor ((,,) a b) | Since: base-4.14.0.0 |
| (Functor f, Functor g) => Functor (Sum f g) | Since: base-4.9.0.0 |
| (Functor f, Functor g) => Functor (f :*: g) | Since: base-4.9.0.0 |
| (Functor f, Functor g) => Functor (f :+: g) | Since: base-4.9.0.0 |
| Functor (K1 i c :: Type -> Type) | Since: base-4.9.0.0 |
| Functor (Cokleisli w a) Source # | |
| Functor f => Functor (WhenMatched f x y) | Since: containers-0.5.9 |
Defined in Data.IntMap.Internal Methods fmap :: (a -> b) -> WhenMatched f x y a -> WhenMatched f x y b # (<$) :: 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 # (<$) :: a -> WhenMissing f k x b -> WhenMissing f k x a # | |
| Functor ((,,,) a b c) | Since: base-4.14.0.0 |
| Functor ((->) r) | Since: base-2.1 |
| (Functor f, Functor g) => Functor (Compose f g) | Since: base-4.9.0.0 |
| (Functor f, Functor g) => Functor (f :.: g) | Since: base-4.9.0.0 |
| Functor f => Functor (M1 i c f) | Since: base-4.9.0.0 |
| Functor f => Functor (WhenMatched f k x y) | Since: containers-0.5.9 |
Defined in Data.Map.Internal Methods fmap :: (a -> b) -> WhenMatched f k x y a -> WhenMatched f k x y b # (<$) :: a -> WhenMatched f k x y b -> WhenMatched f k x y a # | |
| Functor ((,,,,) a b c d) | Since: base-4.18.0.0 |
| Functor ((,,,,,) a b c d e) | Since: base-4.18.0.0 |
| Functor ((,,,,,,) a b c d e f) | Since: base-4.18.0.0 |
(<$>) :: 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 <$.
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