comonad- Haskell 98 comonads

MaintainerEdward Kmett <>







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 andduplicate and the definition of liftW respectively.

class Functor w => Comonad w whereSource



extract :: w a -> aSource

 extract . fmap f = f . extract

duplicate :: w a -> w (w a)Source

 duplicate = extend id
 fmap (fmap f) . duplicate = duplicate . fmap f

extend :: (w a -> b) -> w a -> w bSource

 extend f  = fmap f . duplicate


(=>>) :: Comonad w => w a -> (w a -> b) -> w bSource

extend with the arguments swapped. Dual to >>= for a Monad.

(<<=) :: Comonad w => (w a -> b) -> w a -> w bSource

extend in operator form

liftW :: Comonad w => (a -> b) -> w a -> w bSource

A suitable default definition for fmap for a Comonad. Promotes a function to a comonad.

 fmap f    = extend (f . extract)

wfix :: Comonad w => w (w a -> a) -> aSource

Comonadic fixed point

Cokleisli Arrows

newtype Cokleisli w a b Source

The Cokleisli Arrows of a given Comonad




runCokleisli :: w a -> b

Cokleisli composition

(=>=) :: Comonad w => (w a -> b) -> (w b -> c) -> w a -> cSource

Left-to-right Cokleisli composition

(=<=) :: Comonad w => (w b -> c) -> (w a -> b) -> w a -> cSource

Right-to-left Cokleisli composition