Control.Comonad

Extendable Functors

class Functor w => Extend w where

Methods

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

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

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

 extend f  = fmap f . duplicate

Instances

Extend []

Extend Maybe

Extend Tree

Extend Seq

Extend NonEmpty

Extend Identity

Semigroup m => Extend ((->) m)

Extend (Either a)

Extend ((,) e)

Extend w => Extend (IdentityT w)

(=>=) :: Extend w => (w a -> b) -> (w b -> c) -> w a -> c

Left-to-right Cokleisli composition

(=<=) :: Extend w => (w b -> c) -> (w a -> b) -> w a -> c

Right-to-left Cokleisli composition

(<<=) :: Extend w => (w a -> b) -> w a -> w b

extend in operator form

(=>>) :: Extend w => w a -> (w a -> b) -> w b

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

Comonads

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 Extend w => Comonad w where

$definition

Methods

extract :: w a -> a

 extract . fmap f = f . extract

Instances

Comonad Tree

Comonad NonEmpty

Comonad Identity

(Semigroup m, Monoid m) => Comonad ((->) m)

Comonad ((,) e)

Comonad w => Comonad (IdentityT w)

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

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) -> a

Comonadic fixed point

Cokleisli Arrows

newtype Cokleisli w a b

The Cokleisli Arrows of a given Comonad

Constructors

Cokleisli

Fields

runCokleisli :: w a -> b

Instances

Typeable1 w => Typeable2 (Cokleisli w)

Comonad w => Arrow (Cokleisli w)

Comonad w => ArrowChoice (Cokleisli w)

Comonad w => ArrowApply (Cokleisli w)

Comonad w => Category (Cokleisli w)

Monad (Cokleisli w a)

Functor (Cokleisli w a)

Applicative (Cokleisli w a)