Copyright  (C) 20082015 Edward Kmett (C) 2004 Dave Menendez 

License  BSDstyle (see the file LICENSE) 
Maintainer  Edward Kmett <ekmett@gmail.com> 
Stability  provisional 
Portability  portable 
Safe Haskell  Safe 
Language  Haskell2010 
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:
extend
extract
=id
extract
.extend
f = fextend
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
= 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
=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
idfmap
f =extend
(f .extract
)
These are the default definitions of extend
and duplicate
and
the definition of liftW
respectively.
Instances
Comonad Identity Source #  
Comonad Tree Source #  
Comonad NonEmpty 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 #
Lefttoright Cokleisli
composition
(=<=) :: Comonad w => (w b > c) > (w a > b) > w a > c infixr 1 Source #
Righttoleft 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 semimonoidal comonad on the
category Hask
of Haskell types. That it to say that w
is a strong lax
symmetric semimonoidal 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
Nothing
(<@>) :: 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 Tree Source #  
ComonadApply NonEmpty 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 #
Cokleisli  

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.
See https://www.schoolofhaskell.com/user/edwardk/snippets/fmap or
https://github.com/quchen/articles/blob/master/second_functor_law.md
for an explanation.
fmap :: (a > b) > f a > f b Source #
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
instance that allows
both the last and the penultimate parameters to be mapped over.Bifunctor
Examples
Convert from a
to a Maybe
IntMaybe String
using show
:
>>>
fmap show Nothing
Nothing>>>
fmap show (Just 3)
Just "3"
Convert from an
to 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: base2.1 
Functor Complex  Since: base4.9.0.0 
Functor Identity  Since: base4.8.0.0 
Functor First  Since: base4.8.0.0 
Functor Last  Since: base4.8.0.0 
Functor First  Since: base4.9.0.0 
Functor Last  Since: base4.9.0.0 
Functor Max  Since: base4.9.0.0 
Functor Min  Since: base4.9.0.0 
Functor Dual  Since: base4.8.0.0 
Functor Product  Since: base4.8.0.0 
Functor Sum  Since: base4.8.0.0 
Functor Par1  Since: base4.9.0.0 
Functor P  Since: base4.8.0.0 
Functor ReadP  Since: base2.1 
Functor IntMap  
Functor Digit  
Functor Elem  
Functor FingerTree  
Defined in Data.Sequence.Internal fmap :: (a > b) > FingerTree a > FingerTree b Source # (<$) :: a > FingerTree b > FingerTree a Source #  
Functor Node  
Functor Seq  
Functor ViewL  
Functor ViewR  
Functor Tree  
Functor IO  Since: base2.1 
Functor AnnotDetails  
Defined in Text.PrettyPrint.Annotated.HughesPJ fmap :: (a > b) > AnnotDetails a > AnnotDetails b Source # (<$) :: a > AnnotDetails b > AnnotDetails a Source #  
Functor Doc  
Functor Span  
Functor Q  
Functor TyVarBndr  
Functor NonEmpty  Since: base4.9.0.0 
Functor Maybe  Since: base2.1 
Functor Solo  Since: base4.15 
Functor []  Since: base2.1 
Monad m => Functor (WrappedMonad m)  Since: base2.1 
Defined in Control.Applicative fmap :: (a > b) > WrappedMonad m a > WrappedMonad m b Source # (<$) :: a > WrappedMonad m b > WrappedMonad m a Source #  
Arrow a => Functor (ArrowMonad a)  Since: base4.6.0.0 
Defined in Control.Arrow fmap :: (a0 > b) > ArrowMonad a a0 > ArrowMonad a b Source # (<$) :: a0 > ArrowMonad a b > ArrowMonad a a0 Source #  
Functor (Either a)  Since: base3.0 
Functor (Proxy :: Type > Type)  Since: base4.7.0.0 
Functor (Arg a)  Since: base4.9.0.0 
Functor (Array i)  Since: base2.1 
Functor (U1 :: Type > Type)  Since: base4.9.0.0 
Functor (V1 :: Type > Type)  Since: base4.9.0.0 
Functor (Map k)  
Functor ((,) a)  Since: base2.1 
Arrow a => Functor (WrappedArrow a b)  Since: base2.1 
Defined in Control.Applicative fmap :: (a0 > b0) > WrappedArrow a b a0 > WrappedArrow a b b0 Source # (<$) :: a0 > WrappedArrow a b b0 > WrappedArrow a b a0 Source #  
Functor m => Functor (Kleisli m a)  Since: base4.14.0.0 
Functor (Const m :: Type > Type)  Since: base2.1 
Functor f => Functor (Ap f)  Since: base4.12.0.0 
Functor f => Functor (Alt f)  Since: base4.8.0.0 
(Generic1 f, Functor (Rep1 f)) => Functor (Generically1 f)  Since: base4.17.0.0 
Defined in GHC.Generics fmap :: (a > b) > Generically1 f a > Generically1 f b Source # (<$) :: a > Generically1 f b > Generically1 f a Source #  
Functor f => Functor (Rec1 f)  Since: base4.9.0.0 
Functor (URec (Ptr ()) :: Type > Type)  Since: base4.9.0.0 
Functor (URec Char :: Type > Type)  Since: base4.9.0.0 
Functor (URec Double :: Type > Type)  Since: base4.9.0.0 
Functor (URec Float :: Type > Type)  Since: base4.9.0.0 
Functor (URec Int :: Type > Type)  Since: base4.9.0.0 
Functor (URec Word :: Type > Type)  Since: base4.9.0.0 
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: containers0.5.9 
Defined in Data.IntMap.Internal fmap :: (a > b) > WhenMissing f x a > WhenMissing f x b Source # (<$) :: a > WhenMissing f x b > WhenMissing f x a Source #  
Functor f => Functor (Indexing f)  
Functor (Tagged s)  
Functor m => Functor (IdentityT m)  
Functor ((,,) a b)  Since: base4.14.0.0 
(Functor f, Functor g) => Functor (Sum f g)  Since: base4.9.0.0 
(Functor f, Functor g) => Functor (f :*: g)  Since: base4.9.0.0 
(Functor f, Functor g) => Functor (f :+: g)  Since: base4.9.0.0 
Functor (K1 i c :: Type > Type)  Since: base4.9.0.0 
Functor (Cokleisli w a) Source #  
Functor f => Functor (WhenMatched f x y)  Since: containers0.5.9 
Defined in Data.IntMap.Internal fmap :: (a > b) > WhenMatched f x y a > WhenMatched f x y b Source # (<$) :: a > WhenMatched f x y b > WhenMatched f x y a Source #  
(Applicative f, Monad f) => Functor (WhenMissing f k x)  Since: containers0.5.9 
Defined in Data.Map.Internal fmap :: (a > b) > WhenMissing f k x a > WhenMissing f k x b Source # (<$) :: a > WhenMissing f k x b > WhenMissing f k x a Source #  
Functor ((,,,) a b c)  Since: base4.14.0.0 
Functor ((>) r)  Since: base2.1 
(Functor f, Functor g) => Functor (Compose f g)  Since: base4.9.0.0 
(Functor f, Functor g) => Functor (f :.: g)  Since: base4.9.0.0 
Functor f => Functor (M1 i c f)  Since: base4.9.0.0 
Functor f => Functor (WhenMatched f k x y)  Since: containers0.5.9 
Defined in Data.Map.Internal fmap :: (a > b) > WhenMatched f k x y a > WhenMatched f k x y b Source # (<$) :: 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, <$>
is function
application lifted over a Functor
.
Examples
Convert from a
to a Maybe
Int
using Maybe
String
show
:
>>>
show <$> Nothing
Nothing>>>
show <$> Just 3
Just "3"
Convert from an
to an
Either
Int
Int
Either
Int
String
using show
:
>>>
show <$> Left 17
Left 17>>>
show <$> Right 17
Right "17"
Double each element of a list:
>>>
(*2) <$> [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 <$
.
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: base4.7.0.0