Portability  portable 

Stability  provisional 
Maintainer  Edward Kmett <ekmett@gmail.com> 
Safe Haskell  Trustworthy 
 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
 (=>=) :: 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 where
 (<$>) :: Functor f => (a > b) > f a > f b
 ($>) :: Functor f => f a > b > f b
Comonads
class Functor w => Comonad w whereSource
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.
Combining Comonads
class Comonad w => ComonadApply w whereSource
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
ComonadApply Tree  
ComonadApply NonEmpty  
ComonadApply Identity  
Monoid m => ComonadApply ((>) m)  
Semigroup m => ComonadApply ((,) m)  
ComonadApply w => ComonadApply (IdentityT w) 
(<@@>) :: ComonadApply w => w a > w (a > b) > w bSource
A variant of <@>
with the arguments reversed.
liftW2 :: ComonadApply w => (a > b > c) > w a > w b > w cSource
Lift a binary function into a comonad with zipping
liftW3 :: ComonadApply w => (a > b > c > d) > w a > w b > w c > w dSource
Lift a ternary function into a comonad with zipping
Cokleisli Arrows
newtype Cokleisli w a b Source
Cokleisli  

Typeable1 w => Typeable2 (Cokleisli w)  
Comonad w => Arrow (Cokleisli w)  
Comonad w => ArrowChoice (Cokleisli w)  
Comonad w => ArrowApply (Cokleisli w)  
ComonadApply w => ArrowLoop (Cokleisli w)  
Comonad w => Category (Cokleisli w)  
Monad (Cokleisli w a)  
Functor (Cokleisli w a)  
Applicative (Cokleisli w a) 
Functors
class Functor f where
The Functor
class is used for types that can be mapped over.
Instances of Functor
should satisfy the following laws:
fmap id == id fmap (f . g) == fmap f . fmap g
The instances of Functor
for lists, Maybe
and IO
satisfy these laws.
Functor []  
Functor IO  
Functor ZipList  
Functor Tree  
Functor Option  
Functor NonEmpty  
Functor Identity  
Functor ((>) r)  
Functor (Either a)  
Functor ((,) a)  
Functor (Const m)  
Monad m => Functor (WrappedMonad m)  
Arrow a => Functor (ArrowMonad a)  
Functor m => Functor (IdentityT m)  
Arrow a => Functor (WrappedArrow a b)  
Functor (Tagged k s)  
Functor (Cokleisli w a) 