Copyright (C) 2008-2015 Edward Kmett, (C) 2004 Dave Menendez BSD-style (see the file LICENSE) Edward Kmett provisional portable Safe Haskell2010

Description

Synopsis

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

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

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`.

`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 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 FieldsrunCokleisli :: w a -> b

Instances

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

# 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 ZipList Functor ReadPrec Functor ReadP Functor Maybe Functor Tree Functor Min Functor Max Functor First Functor Last Functor Option Functor NonEmpty Functor Identity 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 (Arg a) Functor m => Functor (IdentityT m) Functor (HashMap k) Arrow a => Functor (WrappedArrow a b) Functor (Tagged k s) (Functor f, Functor g) => Functor (Compose f g) 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 f => (a -> b) -> f a -> f b infixl 4

An infix synonym for `fmap`.

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

Flipped version of `<\$`.

Since: 4.7.0.0
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