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Operator Glossary

One aspect of Haskell that many new users find difficult to get a handle on is operators. Unlike many other languages, Haskell gives a lot of flexibility to developers to define custom operators. This can lead to shorter, more elegant code in many cases. For example, compare these three equivalent pieces of code:

v1 = mappend (mappend "hello " "there ") "world"
v2 = "hello " `mappend` "there " `mappend` "world"
v3 = "hello " ++ "there " ++ "world"

Unfortunately, not all operators are as self-explanatory as the ++ operator (which, in case you're wondering, is "list append"). This page will attempt to cover the most common "surprising" operators. In other words: we won't bother covering common mathematical operators like + or *, nor will we cover operators defined in less common libraries.

Hoogle is your friend

It's worth pointing out as well that for many operators, using the Hoogle search engine can be a great way to find out about an operator, or for that matter any function. It's pretty easy to find ++ that way. Go ahead, try it out now!

Function application $

($) :: (a -> b) -> a -> b

One of the most common operators, and source of initial confusion, is the $ operator. All this does is apply a function. So, f $ x is exactly equivalent to f x. If so, why would you ever use $? The primary reason is - for those who prefer the style - to avoid parentheses. For example, you can replace:

foo (bar (baz bin))


foo $ bar $ baz bin

A less common but arguably more compelling use case is to capture the act of applying a function to an argument. To clarify that rather vague statement with an example:

#!/usr/bin/env stack
-- stack --resolver ghc-7.10.3 runghc

double :: Int -> Int
double x = x + x

square :: Int -> Int
square x = x * x

main :: IO ()
main = print (map ($ 5) [double, square])

The ($ 5) bit means "apply the function to 5", and then we can use map to use it with both the double and square functions.

Function composition .

(.) :: (b -> c) -> (a -> b) -> (a -> c)

Not much more to it than that: take two functions and compose them together.

#!/usr/bin/env stack
-- stack --resolver ghc-7.10.3 runghc

double :: Int -> Int
double x = x + x

square :: Int -> Int
square x = x * x

main :: IO ()
main = (print . double . square) 5

Or you can combine this together with the $ operator to avoid those parentheses if you're so inclined:

main = print . double . square $ 5

In addition to its usage for function composition, the period is also used for hierarchical modules, e.g.:

#!/usr/bin/env stack
-- stack --resolver ghc-7.10.3 runghc
import qualified Data.Monoid

main :: IO ()
main = putStrLn $ Data.Monoid.mappend "hello " "world"

Finally, in the Control.Category module, the Category typeclass also uses the . operator to define categorical composition. This generalizes standard function composition, but is not as commonly used.

Reverse function application &

(&) :: a -> (a -> b) -> b

& is just like $ only backwards. Take our example for $:

foo $ bar $ baz bin

This is semantically equivalent to:

bin & baz & bar & foo

& is useful because the order in which functions are applied to their arguments read left to right instead of the reverse (which is the case for $). This is closer to how English is read so it can improve code clarity.

In our function composition example we composed the functions square, double, and print and applied the resulting function to the number 5.

Rewriting it using & gives us

#!/usr/bin/env stack
-- stack --resolver lts-6.4 runghc
import Data.Function

double :: Int -> Int
double x = x + x

square :: Int -> Int
square x = x * x

main :: IO ()
main = 5 & square & double & print

Monoidal append <>

(<>) :: Monoid m => m -> m -> m

The <> operator is just a synonym for the mappend function. This comes from the Monoid typeclass, which represents types which have an identity and an associative binary operation. Some examples:

  • For lists, <> is the same as ++ (append two lists)
  • For vectors, this logic holds as well
  • For Sets, this is a union operation (all values present in either Set)
  • For Maps, we have a "left biased union", meaning we combine the key/value pairs from both inputs, and if both inputs share a key, the value in the left input is selected
  • For numbers, both addition and multiplication form a Monoid, where 0 is the additive identity (since 0 + x = x) and 1 is the multiplicative identity (since 1 * x = x). Therefore, to avoid confusion, Data.Monoid defines helper newtype wrappers Sum and Product
#!/usr/bin/env stack
-- stack --resolver lts-6.4 runghc
import Data.Monoid ((<>))

main :: IO ()
main = putStrLn $ "hello " <> "there " <> "world!"

Functor map <$>

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

The <$> operator is just a synonym for the fmap function from the Functor typeclass. This function generalizes the map function for lists to many other data types, such as Maybe, IO, and Map.

#!/usr/bin/env stack
-- stack --resolver ghc-7.10.3 runghc
import Data.Monoid ((<>))

main :: IO ()
main = do
    putStrLn "Enter your year of birth"
    year <- read <$> getLine
    let age :: Int
        age = 2020 - year
    putStrLn $ "Age in 2020: " <> show age

In addition, there are two additional operators provided which replace a value inside a Functor instead of applying a function. This can be both more convenient in some cases, as well as for some Functors be more efficient. In terms of definition:

value <$ functor = const value <$> functor
functor $> value = const value <$> functor

x <$ y = y $> x
x $> y = y <$ x

Applicative function application <*>

(<*>) :: Applicative f => f (a -> b) -> f a -> f b
(*>) :: Applicative f => f a -> f b -> f b
(<*) :: Applicative f => f a -> f b -> f a

Commonly seen with <$>, <*> is an operator that applies a wrapped function to a wrapped value. It is part of the Applicative typeclass, and is very often seen in code like the following:

foo <$> bar <*> baz

For cases when you're dealing with a Monad, this is equivalent to:

do x <- bar
   y <- baz
   return (foo x y)

Other common examples including parsers and serialization libraries. Here's an example you might see using the aeson package:

data Person = Person { name :: Text, age :: Int } deriving Show

-- We expect a JSON object, so we fail at any non-Object value.
instance FromJSON Person where
    parseJSON (Object v) = Person <$> v .: "name" <*> v .: "age"
    parseJSON _ = empty

To go along with this, we have two helper operators that are less frequently used:

  • *> ignores the value from the first argument. It can be defined as:

    a1 *> a2 = (id <$ a1) <*> a2

    Or in do-notation:

    a1 *> a2 = do
        _ <- a1

    For Monads, this is completely equivalent to >>.

  • <* is the same thing in reverse: perform the first action then the second, but only take the value from the first action. Again, definitions in terms of <*> and do-notation:

    (<*) = liftA2 const
    a1 <* a2 = do
        res <- a1
        _ <- a2
        return res

Various monadic binding/composition operators

(>>=) :: Monad m => m a -> (a -> m b) -> m b
(=<<) :: Monad m => (a -> m b) -> m a -> m b
(>>) :: Monad m => m a -> m b -> m b
(>=>) :: Monad m => (a -> m b) -> (b -> m c) -> (a -> m c)
(<=<) :: Monad m => (b -> m c) -> (a -> m b) -> (a -> m c)

There are a few different monadic binding operators. The two most basic are >>= and >>, as they can be trivially expressed in do-notation. And as previously mentioned, >> is just a synonym for *> from the Applicative class, so it's even easier. =<< is just >>= with the arguments reversed.

m1 >>= f = do
    x <- m1
    f x

m1 >> m2 = do
    _ <- m1

f =<< m1 = do
    x <- m1
    f x

In addition to these two operators, there are also composition operators for when you have two monadic functions. >=> pipes the result from the left side to the right side, while <=< pipes the result the other way. In other words:

f >=> g = \x -> do
    y <- f x
    g y

g <=< f = \x -> do
    y <- f x
    g y

f >=> g = g <=< f
g >=> f = f <=< g

Alternative <|>

(<|>) :: Alternative f => f a -> f a -> f a

The Alternative typeclass provides a binary operation on applicative functors (<|>), as well as some identity value (empty). This is used in the ecosystem for a number of different activities, e.g.:

  • In parser libraries for defining different alternative parsing options
  • In the async library to run two different Concurrently actions at once and take the first result to succeed
#!/usr/bin/env stack
-- stack --resolver lts-6.4 runghc --package async
import Control.Applicative ((<|>))
import Control.Concurrent (threadDelay)
import Control.Concurrent.Async (Concurrently (..))

main :: IO ()
main = do
    res <- runConcurrently $
        (Concurrently (threadDelay 1000000 >> return (Left "Hello"))) <|>
        (Concurrently (threadDelay 2000000 >> return (Right 42)))
    print res

More operators!

If you're aware of other common operators that cause confusion, please open an issue or a PR to extend this document!