comonad-4.2.7.2: Comonads

Copyright | (C) 2008-2015 Edward Kmett, (C) 2004 Dave Menendez
License | BSD-style (see the file LICENSE)
Maintainer | Edward Kmett <ekmett@gmail.com>
Stability | provisional
Portability | portable
Safe Haskell | Safe
Language | Haskell2010

Control.Comonad

Contents

- Comonads
- Combining Comonads
- Cokleisli Arrows
- Functors

Description

Synopsis

- class Functor w => Comonad w where
  - extract :: w a -> a
  - duplicate :: w a -> w (w a)
  - extend :: (w a -> b) -> w a -> w b
- 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
  - (<@>) :: w (a -> b) -> w a -> w b
  - (@>) :: w a -> w b -> w b
  - (<@) :: w a -> w b -> w a
- (<@@>) :: 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
  - fmap :: (a -> b) -> f a -> f b
  - (<$) :: a -> f b -> f a
- (<$>) :: 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  = 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

extract, (duplicate | extend)

Methods

extract :: w a -> a Source

    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 b Source

    extend f = fmap f . duplicate

Instances

Comonad Identity Source |
Comonad Tree Source |
Comonad NonEmpty Source |
Monoid m => Comonad ((->) m) Source |
Comonad ((,) e) Source |
Comonad (Arg e) Source |
Comonad w => Comonad (IdentityT w) Source |
Comonad (Tagged * s) 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 f, Comonad g) => Comonad (Coproduct f g) Source |

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

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

You can only safely use to define `fmap` if your `Comonad` defined `extend`, not just `duplicate`, since defining `extend` in terms of duplicate uses `fmap`!

    fmap f = liftW f = extend (f . extract)

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

Comonadic fixed point à la David Menendez

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

Comonadic fixed point à la Dominic Orchard

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

Left-to-right `Cokleisli` composition

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

Right-to-left `Cokleisli` composition

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

`extend` in operator form

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

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

# 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 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

Minimal complete definition

Nothing

Methods

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

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

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

Instances

ComonadApply Identity Source |
ComonadApply Tree Source |
ComonadApply NonEmpty Source |
Monoid m => ComonadApply ((->) m) Source |
Semigroup m => ComonadApply ((,) m) Source |
ComonadApply w => ComonadApply (IdentityT w) 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 => 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

The `Cokleisli` `Arrow`s of a given `Comonad`

Constructors

Cokleisli |

Fields

runCokleisli :: w a -> b

Instances

Comonad w => Category * (Cokleisli w) Source |
Comonad w => Arrow (Cokleisli w) Source |
Comonad w => ArrowChoice (Cokleisli w) Source |
Comonad w => ArrowApply (Cokleisli w) Source |
ComonadApply w => ArrowLoop (Cokleisli w) Source |
Monad (Cokleisli w a) Source |
Functor (Cokleisli w a) Source |
Applicative (Cokleisli w a) Source |

# 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.

Minimal complete definition

fmap

Methods

fmap :: (a -> b) -> f a -> f b

(<$) :: a -> f b -> f a infixl 4

Replace all locations in the input with the same value. The default definition is ``fmap` . `const``, but this may be overridden with a more efficient version.

Instances

Functor [] |
Functor IO |
Functor Id |
Functor P |
Functor Identity |
Functor ZipList |
Functor First |
Functor Last |
Functor ReadPrec |
Functor ReadP |
Functor Maybe |
Functor Digit |
Functor Node |
Functor Elem |
Functor FingerTree |
Functor IntMap |
Functor Tree |
Functor Seq |
Functor ViewL |
Functor ViewR |
Functor Min |
Functor Max |
Functor First |
Functor Last |
Functor Option |
Functor NonEmpty |
Functor ((->) r) |
Functor (Either a) |
Functor ((,) a) |
Ix i => Functor (Array i) |
Functor (StateL s) |
Functor (StateR s) |
Functor (Const m) |
Monad m => Functor (WrappedMonad m) |
Arrow a => Functor (ArrowMonad a) |
Functor (Proxy *) |
Functor (State s) |
Functor (Map k) |
Functor (Arg a) |
Functor f => Functor (Reverse f) | Derived instance.
Functor f => Functor (Backwards f) | Derived instance.
Functor m => Functor (MaybeT m) |
Functor m => Functor (ListT m) |
Functor m => Functor (IdentityT m) |
Functor (Constant a) |
Functor (HashMap k) |
Arrow a => Functor (WrappedArrow a b) |
Functor f => Functor (Alt * f) |
Functor (Tagged k s) |
(Functor f, Functor g) => Functor (Sum f g) |
(Functor f, Functor g) => Functor (Product f g) |
(Functor f, Functor g) => Functor (Compose f g) |
Functor m => Functor (WriterT w m) |
Functor m => Functor (WriterT w m) |
Functor m => Functor (ErrorT e m) |
Functor m => Functor (ExceptT e m) |
Functor m => Functor (StateT s m) |
Functor m => Functor (StateT s m) |
Functor m => Functor (ReaderT r m) |
Functor (Cokleisli w a) |
Functor w => Functor (EnvT e w) |
Functor w => Functor (StoreT s w) |
Functor w => Functor (TracedT m w) |
(Functor f, Functor g) => Functor (Coproduct f g) |
Functor m => Functor (RWST r w s m) |
Functor m => Functor (RWST r w s m) |

(<$>) :: Functor f => (a -> b) -> f a -> f b infixl 4

An infix synonym for `fmap`.

#### Examples

Convert from a ``Maybe` `Int`` to a ``Maybe` `String`` using `show`:

    >>> show <$> Nothing
    Nothing
    >>> show <$> Just 3
    Just "3"

Convert from an ``Either` `Int` `Int`` to an ``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

Flipped version of `<$`.

#### Examples

Replace the contents of a ``Maybe` `Int`` with a constant `String`:

    >>> Nothing $> "foo"
    Nothing
    >>> Just 90210 $> "foo"
    Just "foo"

Replace the contents of an ``Either` `Int` `Int`` with a constant `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: 4.7.0.0