planet-mitchell-0.1.0: Planet Mitchell

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Synopsis

class Functor w => Comonad (w :: * -> *) where #

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.

Minimal complete definition

Methods

extract :: w a -> a #

extract . fmap f = f . extract

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

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

Instances

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

Comonadic fixed point à la David Menendez

cfix :: Comonad w => (w a -> a) -> w a #

Comonadic fixed point à la Dominic Orchard

kfix :: ComonadApply w => w (w a -> a) -> w a #

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 #

Left-to-right Cokleisli composition

(=<=) :: Comonad w => (w b -> c) -> (w a -> b) -> w a -> c infixr 1 #

Right-to-left Cokleisli composition

(<<=) :: Comonad w => (w a -> b) -> w a -> w b infixr 1 #

extend in operator form

(=>>) :: Comonad w => w a -> (w a -> b) -> w b infixl 1 #

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

class Comonad w => ComonadApply (w :: * -> *) where #

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:

(.) <$> u <@> v <@> w = u <@> (v <@> w) extract (p <@> q) = extract p (extract q) duplicate (p <@> q) = (<@>) <$> 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

Methods

(<@>) :: w (a -> b) -> w a -> w b infixl 4 #

(@>) :: w a -> w b -> w b infixl 4 #

(<@) :: w a -> w b -> w a infixl 4 #

Instances

(<@@>) :: ComonadApply w => w a -> w (a -> b) -> w b infixl 4 #

A variant of <@> with the arguments reversed.

liftW2 :: ComonadApply w => (a -> b -> c) -> w a -> w b -> w c #

Lift a binary function into a Comonad with zipping

liftW3 :: ComonadApply w => (a -> b -> c -> d) -> w a -> w b -> w c -> w d #

Lift a ternary function into a Comonad with zipping

Newtypes

newtype Cokleisli (w :: k -> *) (a :: k) b :: forall k. (k -> *) -> k -> * -> * #

The Cokleisli Arrows of a given Comonad

Constructors

 Cokleisli FieldsrunCokleisli :: w a -> b
Instances