| 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 ( (,) e) Source # | |
| Comonad ( Arg e) Source # | |
| Comonad ( Tagged s) Source # | |
| Comonad w => Comonad ( IdentityT w) Source # | |
| Comonad w => Comonad ( EnvT e w) Source # | |
| Comonad w => Comonad ( StoreT s w) Source # | |
| ( Comonad w, Monoid m) => Comonad ( TracedT m w) Source # | |
| Monoid m => Comonad ((->) m :: Type -> Type ) Source # | |
| ( Comonad f, Comonad g) => Comonad ( Sum f g) 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 # | |
| ComonadApply w => ComonadApply ( IdentityT w) 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 # | |
| Monoid m => ComonadApply ((->) m :: Type -> Type ) 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 Source #
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 Source #
Using
ApplicativeDo
: '
' can be understood as
the
fmap
f as
do
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 Source # (<$) :: a -> FingerTree b -> FingerTree a Source # |
|
| Functor Digit | |
| Functor Node | |
| Functor Elem | |
| Functor ViewL | |
| Functor ViewR | |
| Functor P |
Since: base-4.8.0.0 |
| Functor ( Either a) |
Since: base-3.0 |
| Functor ( V1 :: Type -> Type ) |
Since: base-4.9.0.0 |
| Functor ( U1 :: Type -> Type ) |
Since: base-4.9.0.0 |
| Functor ( (,) a) |
Since: base-2.1 |
| Functor ( Array i) |
Since: base-2.1 |
| Functor ( Arg a) |
Since: base-4.9.0.0 |
| Monad m => Functor ( WrappedMonad m) |
Since: base-2.1 |
|
Defined in Control.Applicative Methods fmap :: (a -> b) -> WrappedMonad m a -> WrappedMonad m b Source # (<$) :: a -> WrappedMonad m b -> WrappedMonad m a Source # |
|
| 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 Source # (<$) :: a0 -> ArrowMonad a b -> ArrowMonad a a0 Source # |
|
| Functor ( Proxy :: Type -> Type ) |
Since: base-4.7.0.0 |
| Functor ( Map k) | |
| Functor f => Functor ( Rec1 f) |
Since: base-4.9.0.0 |
| Functor ( URec Char :: Type -> Type ) |
Since: base-4.9.0.0 |
| Functor ( URec Double :: Type -> Type ) |
Since: base-4.9.0.0 |
| Functor ( URec Float :: Type -> Type ) |
Since: base-4.9.0.0 |
| Functor ( URec Int :: Type -> Type ) |
Since: base-4.9.0.0 |
| Functor ( URec Word :: Type -> Type ) |
Since: base-4.9.0.0 |
| Functor ( URec ( Ptr ()) :: Type -> Type ) |
Since: base-4.9.0.0 |
| Functor ( (,,) a b) |
Since: base-4.14.0.0 |
| Arrow a => Functor ( WrappedArrow a b) |
Since: base-2.1 |
|
Defined in Control.Applicative Methods fmap :: (a0 -> b0) -> WrappedArrow a b a0 -> WrappedArrow a b b0 Source # (<$) :: a0 -> WrappedArrow a b b0 -> WrappedArrow a b a0 Source # |
|
| 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 Source # (<$) :: a -> WhenMissing f x b -> WhenMissing f x a Source # |
|
| Functor f => Functor (Indexing f) | |
| Functor ( Tagged s) | |
| Functor f => Functor ( Reverse f) |
Derived instance. |
| Functor ( Constant a :: Type -> Type ) | |
| Functor m => Functor ( ReaderT r m) | |
| Functor m => Functor ( IdentityT m) | |
| Functor f => Functor ( Backwards f) |
Derived instance. |
| Functor w => Functor ( EnvT e w) Source # | |
| Functor w => Functor ( StoreT s w) Source # | |
| Functor w => Functor ( TracedT m w) Source # | |
| Functor ((->) r :: Type -> Type ) |
Since: base-2.1 |
| Functor ( K1 i c :: Type -> Type ) |
Since: base-4.9.0.0 |
| ( Functor f, Functor g) => Functor (f :+: g) |
Since: base-4.9.0.0 |
| ( Functor f, Functor g) => Functor (f :*: g) |
Since: base-4.9.0.0 |
| Functor ( (,,,) a b c) |
Since: base-4.14.0.0 |
| ( Functor f, Functor g) => Functor ( Product f g) |
Since: base-4.9.0.0 |
| ( Functor f, Functor g) => Functor ( Sum f g) |
Since: base-4.9.0.0 |
| Functor f => Functor ( WhenMatched f x y) |
Since: containers-0.5.9 |
|
Defined in Data.IntMap.Internal Methods fmap :: (a -> b) -> WhenMatched f x y a -> WhenMatched f x y b Source # (<$) :: a -> WhenMatched f x y b -> WhenMatched f x y a Source # |
|
| ( 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 Source # (<$) :: a -> WhenMissing f k x b -> WhenMissing f k x a Source # |
|
| Functor ( Cokleisli w a) Source # | |
| Functor f => Functor ( M1 i c f) |
Since: base-4.9.0.0 |
| ( Functor f, Functor g) => Functor (f :.: g) |
Since: base-4.9.0.0 |
| ( Functor f, Functor g) => Functor ( Compose f g) |
Since: base-4.9.0.0 |
| Functor f => Functor ( WhenMatched f k x y) |
Since: containers-0.5.9 |
|
Defined in Data.Map.Internal Methods fmap :: (a -> b) -> WhenMatched f k x y a -> WhenMatched f k x y b Source # (<$) :: a -> WhenMatched f k x y b -> WhenMatched f k x y a Source # |
|
(<$>) :: Functor f => (a -> b) -> f a -> f b infixl 4 Source #
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,
<$>
is function
application lifted over a
Functor
.
Examples
Convert from a
to a
Maybe
Int
using
Maybe
String
show
:
>>>show <$> NothingNothing>>>show <$> Just 3Just "3"
Convert from an
to an
Either
Int
Int
Either
Int
String
using
show
:
>>>show <$> 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 Source #
Flipped version of
<$
.
Using
ApplicativeDo
: '
as
' can be understood as the
$>
b
do
expression
do as pure b
with an inferred
Functor
constraint.
Examples
Replace the contents of a
with a constant
Maybe
Int
String
:
>>>Nothing $> "foo"Nothing>>>Just 90210 $> "foo"Just "foo"
Replace the contents of an
with a constant
Either
Int
Int
String
, 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