comonad-5: 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
- 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
- (<@@>) :: 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 #

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 #

    extract . fmap f = f . extract

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

Comonad Identity # |
Methods extract :: Identity a -> a # duplicate :: Identity a -> Identity (Identity a) # extend :: (Identity a -> b) -> Identity a -> Identity b #
Comonad NonEmpty # |
Methods extract :: NonEmpty a -> a # duplicate :: NonEmpty a -> NonEmpty (NonEmpty a) # extend :: (NonEmpty a -> b) -> NonEmpty a -> NonEmpty b #
Comonad Tree # |
Methods extract :: Tree a -> a # duplicate :: Tree a -> Tree (Tree a) # extend :: (Tree a -> b) -> Tree a -> Tree b #
Monoid m => Comonad ((->) m) # |
Methods extract :: (m -> a) -> a # duplicate :: (m -> a) -> m -> m -> a # extend :: ((m -> a) -> b) -> (m -> a) -> m -> b #
Comonad ((,) e) # |
Methods extract :: (e, a) -> a # duplicate :: (e, a) -> (e, (e, a)) # extend :: ((e, a) -> b) -> (e, a) -> (e, b) #
Comonad (Arg e) # |
Methods extract :: Arg e a -> a # duplicate :: Arg e a -> Arg e (Arg e a) # extend :: (Arg e a -> b) -> Arg e a -> Arg e b #
Comonad (Tagged * s) # |
Methods extract :: Tagged * s a -> a # duplicate :: Tagged * s a -> Tagged * s (Tagged * s a) # extend :: (Tagged * s a -> b) -> Tagged * s a -> Tagged * s b #
Comonad w => Comonad (IdentityT * w) # |
Methods extract :: IdentityT * w a -> a # duplicate :: IdentityT * w a -> IdentityT * w (IdentityT * w a) # extend :: (IdentityT * w a -> b) -> IdentityT * w a -> IdentityT * w b #
Comonad w => Comonad (EnvT e w) # |
Methods extract :: EnvT e w a -> a # duplicate :: EnvT e w a -> EnvT e w (EnvT e w a) # extend :: (EnvT e w a -> b) -> EnvT e w a -> EnvT e w b #
Comonad w => Comonad (StoreT s w) # |
Methods extract :: StoreT s w a -> a # duplicate :: StoreT s w a -> StoreT s w (StoreT s w a) # extend :: (StoreT s w a -> b) -> StoreT s w a -> StoreT s w b #
(Comonad w, Monoid m) => Comonad (TracedT m w) # |
Methods extract :: TracedT m w a -> a # duplicate :: TracedT m w a -> TracedT m w (TracedT m w a) # extend :: (TracedT m w a -> b) -> TracedT m w a -> TracedT m w b #
(Comonad f, Comonad g) => Comonad (Sum * f g) # |
Methods extract :: Sum * f g a -> a # duplicate :: Sum * f g a -> Sum * f g (Sum * f g a) # extend :: (Sum * f g a -> b) -> Sum * f g a -> Sum * f g b #

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

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 #

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`.

# Combining Comonads

class Comonad w => ComonadApply w where #

`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

Methods

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

(<@>) :: Applicative w => 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 Identity # |
Methods (<@>) :: Identity (a -> b) -> Identity a -> Identity b # (@>) :: Identity a -> Identity b -> Identity b # (<@) :: Identity a -> Identity b -> Identity a #
ComonadApply NonEmpty # |
Methods (<@>) :: NonEmpty (a -> b) -> NonEmpty a -> NonEmpty b # (@>) :: NonEmpty a -> NonEmpty b -> NonEmpty b # (<@) :: NonEmpty a -> NonEmpty b -> NonEmpty a #
ComonadApply Tree # |
Methods (<@>) :: Tree (a -> b) -> Tree a -> Tree b # (@>) :: Tree a -> Tree b -> Tree b # (<@) :: Tree a -> Tree b -> Tree a #
Monoid m => ComonadApply ((->) m) # |
Methods (<@>) :: (m -> a -> b) -> (m -> a) -> m -> b # (@>) :: (m -> a) -> (m -> b) -> m -> b # (<@) :: (m -> a) -> (m -> b) -> m -> a #
Semigroup m => ComonadApply ((,) m) # |
Methods (<@>) :: (m, a -> b) -> (m, a) -> (m, b) # (@>) :: (m, a) -> (m, b) -> (m, b) # (<@) :: (m, a) -> (m, b) -> (m, a) #
ComonadApply w => ComonadApply (IdentityT * w) # |
Methods (<@>) :: IdentityT * w (a -> b) -> IdentityT * w a -> IdentityT * w b # (@>) :: IdentityT * w a -> IdentityT * w b -> IdentityT * w b # (<@) :: IdentityT * w a -> IdentityT * w b -> IdentityT * w a #
(Semigroup e, ComonadApply w) => ComonadApply (EnvT e w) # |
Methods (<@>) :: EnvT e w (a -> b) -> EnvT e w a -> EnvT e w b # (@>) :: EnvT e w a -> EnvT e w b -> EnvT e w b # (<@) :: EnvT e w a -> EnvT e w b -> EnvT e w a #
(ComonadApply w, Semigroup s) => ComonadApply (StoreT s w) # |
Methods (<@>) :: StoreT s w (a -> b) -> StoreT s w a -> StoreT s w b # (@>) :: StoreT s w a -> StoreT s w b -> StoreT s w b # (<@) :: StoreT s w a -> StoreT s w b -> StoreT s w a #
(ComonadApply w, Monoid m) => ComonadApply (TracedT m w) # |
Methods (<@>) :: TracedT m w (a -> b) -> TracedT m w a -> TracedT m w b # (@>) :: TracedT m w a -> TracedT m w b -> TracedT m w b # (<@) :: TracedT m w a -> TracedT m w b -> TracedT m w a #

(<@@>) :: 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

# Cokleisli Arrows

newtype Cokleisli w a b #

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

Constructors

Cokleisli |
Fields runCokleisli :: w a -> b

Instances

Comonad w => Arrow (Cokleisli w) # |
Methods arr :: (b -> c) -> Cokleisli w b c # first :: Cokleisli w b c -> Cokleisli w (b, d) (c, d) # second :: Cokleisli w b c -> Cokleisli w (d, b) (d, c) # (***) :: Cokleisli w b c -> Cokleisli w b' c' -> Cokleisli w (b, b') (c, c') # (&&&) :: Cokleisli w b c -> Cokleisli w b c' -> Cokleisli w b (c, c') #
Comonad w => ArrowChoice (Cokleisli w) # |
Methods left :: Cokleisli w b c -> Cokleisli w (Either b d) (Either c d) # right :: Cokleisli w b c -> Cokleisli w (Either d b) (Either d c) # (+++) :: Cokleisli w b c -> Cokleisli w b' c' -> Cokleisli w (Either b b') (Either c c') # (|||) :: Cokleisli w b d -> Cokleisli w c d -> Cokleisli w (Either b c) d #
Comonad w => ArrowApply (Cokleisli w) # |
Methods app :: Cokleisli w (Cokleisli w b c, b) c #
ComonadApply w => ArrowLoop (Cokleisli w) # |
Methods loop :: Cokleisli w (b, d) (c, d) -> Cokleisli w b c #
Comonad w => Category * (Cokleisli w) # |
Methods id :: cat a a # (.) :: cat b c -> cat a b -> cat a c #
Monad (Cokleisli w a) # |
Methods (>>=) :: Cokleisli w a a -> (a -> Cokleisli w a b) -> Cokleisli w a b # (>>) :: Cokleisli w a a -> Cokleisli w a b -> Cokleisli w a b # return :: a -> Cokleisli w a a # fail :: String -> Cokleisli w a a #
Functor (Cokleisli w a) # |
Methods fmap :: (a -> b) -> Cokleisli w a a -> Cokleisli w a b # (<$) :: a -> Cokleisli w a b -> Cokleisli w a a #
Applicative (Cokleisli w a) # |
Methods pure :: a -> Cokleisli w a a # (<*>) :: Cokleisli w a (a -> b) -> Cokleisli w a a -> Cokleisli w a b # (*>) :: Cokleisli w a a -> Cokleisli w a b -> Cokleisli w a b # (<*) :: Cokleisli w a a -> Cokleisli w a b -> Cokleisli w a a #

# 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

Instances

Functor [] |
Methods fmap :: (a -> b) -> [a] -> [b] # (<$) :: a -> [b] -> [a] #
Functor Maybe |
Methods fmap :: (a -> b) -> Maybe a -> Maybe b # (<$) :: a -> Maybe b -> Maybe a #
Functor IO |
Methods fmap :: (a -> b) -> IO a -> IO b # (<$) :: a -> IO b -> IO a #
Functor V1 |
Methods fmap :: (a -> b) -> V1 a -> V1 b # (<$) :: a -> V1 b -> V1 a #
Functor U1 |
Methods fmap :: (a -> b) -> U1 a -> U1 b # (<$) :: a -> U1 b -> U1 a #
Functor Par1 |
Methods fmap :: (a -> b) -> Par1 a -> Par1 b # (<$) :: a -> Par1 b -> Par1 a #
Functor Id |
Methods fmap :: (a -> b) -> Id a -> Id b # (<$) :: a -> Id b -> Id a #
Functor Identity |
Methods fmap :: (a -> b) -> Identity a -> Identity b # (<$) :: a -> Identity b -> Identity a #
Functor Min |
Methods fmap :: (a -> b) -> Min a -> Min b # (<$) :: a -> Min b -> Min a #
Functor Max |
Methods fmap :: (a -> b) -> Max a -> Max b # (<$) :: a -> Max b -> Max a #
Functor First |
Methods fmap :: (a -> b) -> First a -> First b # (<$) :: a -> First b -> First a #
Functor Last |
Methods fmap :: (a -> b) -> Last a -> Last b # (<$) :: a -> Last b -> Last a #
Functor Option |
Methods fmap :: (a -> b) -> Option a -> Option b # (<$) :: a -> Option b -> Option a #
Functor NonEmpty |
Methods fmap :: (a -> b) -> NonEmpty a -> NonEmpty b # (<$) :: a -> NonEmpty b -> NonEmpty a #
Functor Complex |
Methods fmap :: (a -> b) -> Complex a -> Complex b # (<$) :: a -> Complex b -> Complex a #
Functor ZipList |
Methods fmap :: (a -> b) -> ZipList a -> ZipList b # (<$) :: a -> ZipList b -> ZipList a #
Functor Dual |
Methods fmap :: (a -> b) -> Dual a -> Dual b # (<$) :: a -> Dual b -> Dual a #
Functor Sum |
Methods fmap :: (a -> b) -> Sum a -> Sum b # (<$) :: a -> Sum b -> Sum a #
Functor Product |
Methods fmap :: (a -> b) -> Product a -> Product b # (<$) :: a -> Product b -> Product a #
Functor First |
Methods fmap :: (a -> b) -> First a -> First b # (<$) :: a -> First b -> First a #
Functor Last |
Methods fmap :: (a -> b) -> Last a -> Last b # (<$) :: a -> Last b -> Last a #
Functor Digit |
Methods fmap :: (a -> b) -> Digit a -> Digit b # (<$) :: a -> Digit b -> Digit a #
Functor Node |
Methods fmap :: (a -> b) -> Node a -> Node b # (<$) :: a -> Node b -> Node a #
Functor Elem |
Methods fmap :: (a -> b) -> Elem a -> Elem b # (<$) :: a -> Elem b -> Elem a #
Functor FingerTree |
Methods fmap :: (a -> b) -> FingerTree a -> FingerTree b # (<$) :: a -> FingerTree b -> FingerTree a #
Functor Tree |
Methods fmap :: (a -> b) -> Tree a -> Tree b # (<$) :: a -> Tree b -> Tree a #
Functor Seq |
Methods fmap :: (a -> b) -> Seq a -> Seq b # (<$) :: a -> Seq b -> Seq a #
Functor ViewL |
Methods fmap :: (a -> b) -> ViewL a -> ViewL b # (<$) :: a -> ViewL b -> ViewL a #
Functor ViewR |
Methods fmap :: (a -> b) -> ViewR a -> ViewR b # (<$) :: a -> ViewR b -> ViewR a #
Functor ((->) r) |
Methods fmap :: (a -> b) -> (r -> a) -> r -> b # (<$) :: a -> (r -> b) -> r -> a #
Functor (Either a) |
Methods fmap :: (a -> b) -> Either a a -> Either a b # (<$) :: a -> Either a b -> Either a a #
Functor f => Functor (Rec1 f) |
Methods fmap :: (a -> b) -> Rec1 f a -> Rec1 f b # (<$) :: a -> Rec1 f b -> Rec1 f a #
Functor (URec Char) |
Methods fmap :: (a -> b) -> URec Char a -> URec Char b # (<$) :: a -> URec Char b -> URec Char a #
Functor (URec Double) |
Methods fmap :: (a -> b) -> URec Double a -> URec Double b # (<$) :: a -> URec Double b -> URec Double a #
Functor (URec Float) |
Methods fmap :: (a -> b) -> URec Float a -> URec Float b # (<$) :: a -> URec Float b -> URec Float a #
Functor (URec Int) |
Methods fmap :: (a -> b) -> URec Int a -> URec Int b # (<$) :: a -> URec Int b -> URec Int a #
Functor (URec Word) |
Methods fmap :: (a -> b) -> URec Word a -> URec Word b # (<$) :: a -> URec Word b -> URec Word a #
Functor (URec (Ptr ())) |
Methods fmap :: (a -> b) -> URec (Ptr ()) a -> URec (Ptr ()) b # (<$) :: a -> URec (Ptr ()) b -> URec (Ptr ()) a #
Functor ((,) a) |
Methods fmap :: (a -> b) -> (a, a) -> (a, b) # (<$) :: a -> (a, b) -> (a, a) #
Functor (Array i) |
Methods fmap :: (a -> b) -> Array i a -> Array i b # (<$) :: a -> Array i b -> Array i a #
Functor (StateL s) |
Methods fmap :: (a -> b) -> StateL s a -> StateL s b # (<$) :: a -> StateL s b -> StateL s a #
Functor (StateR s) |
Methods fmap :: (a -> b) -> StateR s a -> StateR s b # (<$) :: a -> StateR s b -> StateR s a #
Functor (Arg a) |
Methods fmap :: (a -> b) -> Arg a a -> Arg a b # (<$) :: a -> Arg a b -> Arg a a #
Monad m => Functor (WrappedMonad m) |
Methods fmap :: (a -> b) -> WrappedMonad m a -> WrappedMonad m b # (<$) :: a -> WrappedMonad m b -> WrappedMonad m a #
Arrow a => Functor (ArrowMonad a) |
Methods fmap :: (a -> b) -> ArrowMonad a a -> ArrowMonad a b # (<$) :: a -> ArrowMonad a b -> ArrowMonad a a #
Functor (Proxy *) |
Methods fmap :: (a -> b) -> Proxy * a -> Proxy * b # (<$) :: a -> Proxy * b -> Proxy * a #
Functor (State s) |
Methods fmap :: (a -> b) -> State s a -> State s b # (<$) :: a -> State s b -> State s a #
Functor (K1 i c) |
Methods fmap :: (a -> b) -> K1 i c a -> K1 i c b # (<$) :: a -> K1 i c b -> K1 i c a #
(Functor f, Functor g) => Functor ((:+:) f g) |
Methods fmap :: (a -> b) -> (f :+: g) a -> (f :+: g) b # (<$) :: a -> (f :+: g) b -> (f :+: g) a #
(Functor f, Functor g) => Functor ((:*:) f g) |
Methods fmap :: (a -> b) -> (f :*: g) a -> (f :*: g) b # (<$) :: a -> (f :*: g) b -> (f :*: g) a #
(Functor f, Functor g) => Functor ((:.:) f g) |
Methods fmap :: (a -> b) -> (f :.: g) a -> (f :.: g) b # (<$) :: a -> (f :.: g) b -> (f :.: g) a #
Arrow a => Functor (WrappedArrow a b) |
Methods fmap :: (a -> b) -> WrappedArrow a b a -> WrappedArrow a b b # (<$) :: a -> WrappedArrow a b b -> WrappedArrow a b a #
Functor (Const * m) |
Methods fmap :: (a -> b) -> Const * m a -> Const * m b # (<$) :: a -> Const * m b -> Const * m a #
Functor f => Functor (Alt * f) |
Methods fmap :: (a -> b) -> Alt * f a -> Alt * f b # (<$) :: a -> Alt * f b -> Alt * f a #
Functor (Tagged k s) |
Methods fmap :: (a -> b) -> Tagged k s a -> Tagged k s b # (<$) :: a -> Tagged k s b -> Tagged k s a #
Functor f => Functor (Reverse * f) | Derived instance.
Methods fmap :: (a -> b) -> Reverse * f a -> Reverse * f b # (<$) :: a -> Reverse * f b -> Reverse * f a #
Functor f => Functor (Backwards * f) | Derived instance.
Methods fmap :: (a -> b) -> Backwards * f a -> Backwards * f b # (<$) :: a -> Backwards * f b -> Backwards * f a #
Functor m => Functor (IdentityT * m) |
Methods fmap :: (a -> b) -> IdentityT * m a -> IdentityT * m b # (<$) :: a -> IdentityT * m b -> IdentityT * m a #
Functor (Cokleisli w a) # |
Methods fmap :: (a -> b) -> Cokleisli w a a -> Cokleisli w a b # (<$) :: a -> Cokleisli w a b -> Cokleisli w a a #
Functor w => Functor (EnvT e w) # |
Methods fmap :: (a -> b) -> EnvT e w a -> EnvT e w b # (<$) :: a -> EnvT e w b -> EnvT e w a #
Functor w => Functor (StoreT s w) # |
Methods fmap :: (a -> b) -> StoreT s w a -> StoreT s w b # (<$) :: a -> StoreT s w b -> StoreT s w a #
Functor w => Functor (TracedT m w) # |
Methods fmap :: (a -> b) -> TracedT m w a -> TracedT m w b # (<$) :: a -> TracedT m w b -> TracedT m w a #
Functor f => Functor (M1 i c f) |
Methods fmap :: (a -> b) -> M1 i c f a -> M1 i c f b # (<$) :: a -> M1 i c f b -> M1 i c f a #
(Functor f, Functor g) => Functor (Sum * f g) |
Methods fmap :: (a -> b) -> Sum * f g a -> Sum * f g b # (<$) :: a -> Sum * f g b -> Sum * f g a #
(Functor f, Functor g) => Functor (Product * f g) |
Methods fmap :: (a -> b) -> Product * f g a -> Product * f g b # (<$) :: a -> Product * f g b -> Product * f g a #
Functor m => Functor (ReaderT * r m) |
Methods fmap :: (a -> b) -> ReaderT * r m a -> ReaderT * r m b # (<$) :: a -> ReaderT * r m b -> ReaderT * r m a #
(Functor f, Functor g) => Functor (Compose * * f g) |
Methods fmap :: (a -> b) -> Compose * * f g a -> Compose * * f g b # (<$) :: a -> Compose * * f g b -> Compose * * f g a #

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

An infix synonym for `fmap`.

The name of this operator is an allusion to `$`. Note the similarities between their types:

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

Whereas `$` is function application, `<$>` is function application lifted over a `Functor`.

#### 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