Class of data structures that can be traversed from left to right, performing an action on each element. Instances are expected to satisfy the listed laws.

Traversable structures support element-wise sequencing of Applicative effects (thus also Monad effects) to construct new structures of the same shape as the input.

To illustrate what is meant by same shape, if the input structure is [a], each output structure is a list [b] of the same length as the input. If the input is a Tree a, each output Tree b has the same graph of intermediate nodes and leaves. Similarly, if the input is a 2-tuple (x, a), each output is a 2-tuple (x, b), and so forth.

It is in fact possible to decompose a traversable structure t a into its shape (a.k.a. spine) of type t () and its element list [a]. The original structure can be faithfully reconstructed from its spine and element list.

The implementation of a Traversable instance for a given structure follows naturally from its type; see the Construction section for details. Instances must satisfy the laws listed in the Laws section. The diverse uses of Traversable structures result from the many possible choices of Applicative effects. See the Advanced Traversals section for some examples.

Every Traversable structure is both a Functor and Foldable because it is possible to implement the requisite instances in terms of traverse by using fmapDefault for fmap and foldMapDefault for foldMap. Direct fine-tuned implementations of these superclass methods can in some cases be more efficient.

For an Applicative functor f and a Traversable functor t, the type signatures of traverse and fmap are rather similar:

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

The key difference is that fmap produces a structure whose elements (of type f b) are individual effects, while traverse produces an aggregate effect yielding structures of type t b.

For example, when f is the IO monad, and t is List, fmap yields a list of IO actions, whereas traverse constructs an IO action that evaluates to a list of the return values of the individual actions performed left-to-right.

traverse :: (a -> IO b) -> [a] -> IO [b]

The mapM function is a specialisation of traverse to the case when f is a Monad. For monads, mapM is more idiomatic than traverse. The two are otherwise generally identical (though mapM may be specifically optimised for monads, and could be more efficient than using the more general traverse).

traverse :: (Applicative f, Traversable t) => (a -> f b) -> t a -> f (t b)
mapM     :: (Monad       m, Traversable t) => (a -> m b) -> t a -> m (t b)

When the traversable term is a simple variable or expression, and the monadic action to run is a non-trivial do block, it can be more natural to write the action last. This idiom is supported by for, forM, and forAccumM which are the flipped versions of traverse, mapM, and mapAccumM respectively.

The traverse and mapM methods have analogues in the Data.Foldable module. These are traverse_ and mapM_, and their flipped variants for_ and forM_, respectively. The result type is f (), they don't return an updated structure, and can be used to sequence effects over all the elements of a Traversable (any Foldable) structure just for their side-effects.

If the Traversable structure is empty, the result is pure (). When effects short-circuit, the f () result may, for example, be Nothing if f is Maybe, or Left e when it is Either e.

It is perhaps worth noting that Maybe is not only a potential Applicative functor for the return value of the first argument of traverse, but is also itself a Traversable structure with either zero or one element. A convenient idiom for conditionally executing an action just for its effects on a Just value, and doing nothing otherwise is:

-- action :: Monad m => a -> m ()
-- mvalue :: Maybe a
mapM_ action mvalue -- :: m ()

which is more concise than:

maybe (return ()) action mvalue

The mapM_ idiom works verbatim if the type of mvalue is later refactored from Maybe a to Either e a (assuming it remains OK to silently do nothing in the Left case).

When traverse or mapM is applied to an empty structure ts (one for which null ts is True) the return value is pure ts regardless of the provided function g :: a -> f b. It is not possible to apply the function when no values of type a are available, but its type determines the relevant instance of pure.

null ts ==> traverse g ts == pure ts

Otherwise, when ts is non-empty and at least one value of type b results from each f a, the structures t b have the same shape (list length, graph of tree nodes, ...) as the input structure t a, but the slots previously occupied by elements of type a now hold elements of type b.

A single traversal may produce one, zero or many such structures. The zero case happens when one of the effects f a sequenced as part of the traversal yields no replacement values. Otherwise, the many case happens when one of sequenced effects yields multiple values.

The traverse function does not perform selective filtering of slots in the output structure as with e.g. mapMaybe.

>>> let incOdd n = if odd n then Just $ n + 1 else Nothing
>>> mapMaybe incOdd [1, 2, 3]
[2,4]
>>> traverse incOdd [1, 3, 5]
Just [2,4,6]
>>> traverse incOdd [1, 2, 3]
Nothing

In the above examples, with Maybe as the Applicative f, we see that the number of t b structures produced by traverse may differ from one: it is zero when the result short-circuits to Nothing. The same can happen when f is List and the result is [], or f is Either e and the result is Left (x :: e), or perhaps the empty value of some Alternative functor.

When f is e.g. List, and the map g :: a -> [b] returns more than one value for some inputs a (and at least one for all a), the result of mapM g ts will contain multiple structures of the same shape as ts:

List.length (mapM g ts) == List.product (fmap (List.length . g) ts)

For example:

>>> List.length $ mapM (\n -> [1..n]) [1..6]
720
>>> List.product $ List.length . (\n -> [1..n]) <$> [1..6]
720

In other words, a traversal with a function g :: a -> [b], over an input structure t a, yields a list [t b], whose length is the product of the lengths of the lists that g returns for each element of the input structure! The individual elements a of the structure are replaced by each element of g a in turn:

>>> mapM (\n -> [1..n]) $ Just 3
[Just 1,Just 2,Just 3]
>>> mapM (\n -> [1..n]) [1..3]
[[1,1,1],[1,1,2],[1,1,3],[1,2,1],[1,2,2],[1,2,3]]

If any element of the structure t a is mapped by g to an empty list, then the entire aggregate result is empty, because no value is available to fill one of the slots of the output structure:

>>> mapM (\n -> [1..n]) $ [0..6] -- [1..0] is empty
[]

The sequenceA and sequence methods are useful when what you have is a container of pending applicative or monadic effects, and you want to combine them into a single effect that produces zero or more containers with the computed values.

sequenceA :: (Applicative f, Traversable t) => t (f a) -> f (t a)
sequence  :: (Monad       m, Traversable t) => t (m a) -> m (t a)
sequenceA = traverse id -- default definition
sequence  = sequenceA   -- default definition

When the monad m is IO, applying sequence to a list of IO actions, performs each in turn, returning a list of the results:

sequence [putStr "Hello ", putStrLn "World!"]
    = (\a b -> [a,b]) <$> putStr "Hello " <*> putStrLn "World!"
    = do u1 <- putStr "Hello "
         u2 <- putStrLn "World!"
         return [u1, u2]         -- In this case  [(), ()]

For sequenceA, the non-deterministic behaviour of List is most easily seen in the case of a list of lists (of elements of some common fixed type). The result is a cross-product of all the sublists:

>>> sequenceA [[0, 1, 2], [30, 40], [500]]
[[0,30,500],[0,40,500],[1,30,500],[1,40,500],[2,30,500],[2,40,500]]

Because the input list has three (sublist) elements, the result is a list of triples (same shape).

The traverse method has a default implementation in terms of sequenceA:

traverse g = sequenceA . fmap g

but relying on this default implementation is not recommended, it requires that the structure is already independently a Functor. The definition of sequenceA in terms of traverse id is much simpler than traverse expressed via a composition of sequenceA and fmap. Instances should generally implement traverse explicitly. It may in some cases also make sense to implement a specialised mapM.

Because fmapDefault is defined in terms of traverse (whose default definition in terms of sequenceA uses fmap), you must not use fmapDefault to define the Functor instance if the Traversable instance directly defines only sequenceA.

When the monad m is Either or Maybe (more generally any MonadPlus), the effect in question is to short-circuit the result on encountering Left or Nothing (more generally mzero).

>>> sequence [Just 1,Just 2,Just 3]
Just [1,2,3]
>>> sequence [Just 1,Nothing,Just 3]
Nothing
>>> sequence [Right 1,Right 2,Right 3]
Right [1,2,3]
>>> sequence [Right 1,Left "sorry",Right 3]
Left "sorry"

The result of sequence is all-or-nothing, either structures of exactly the same shape as the input or none at all. The sequence function does not perform selective filtering as with e.g. catMaybes or rights:

>>> catMaybes [Just 1,Nothing,Just 3]
[1,3]
>>> rights [Right 1,Left "sorry",Right 3]
[1,3]

The definition of a Traversable instance for a binary tree is rather similar to the corresponding instance of Functor, given the data type:

data Tree a = Empty | Leaf a | Node (Tree a) a (Tree a)

a canonical Functor instance would be

instance Functor Tree where
   fmap g Empty        = Empty
   fmap g (Leaf x)     = Leaf (g x)
   fmap g (Node l k r) = Node (fmap g l) (g k) (fmap g r)

a canonical Traversable instance would be

instance Traversable Tree where
   traverse g Empty        = pure Empty
   traverse g (Leaf x)     = Leaf <$> g x
   traverse g (Node l k r) = Node <$> traverse g l <*> g k <*> traverse g r

This definition works for any g :: a -> f b, with f an Applicative functor, as the laws for (<*>) imply the requisite associativity.

We can add an explicit non-default mapM if desired:

   mapM g Empty        = return Empty
   mapM g (Leaf x)     = Leaf <$> g x
   mapM g (Node l k r) = do
       ml <- mapM g l
       mk <- g k
       mr <- mapM g r
       return $ Node ml mk mr

See Construction below for a more detailed exploration of the general case, but as mentioned in Overview above, instance definitions are typically rather simple, all the interesting behaviour is a result of an interesting choice of Applicative functor for a traversal.

It is perhaps worth noting that the traversal defined above gives an in-order sequencing of the elements. If instead you want either pre-order (parent first, then child nodes) or post-order (child nodes first, then parent) sequencing, you can define the instance accordingly:

inOrderNode :: Tree a -> a -> Tree a -> Tree a
inOrderNode l x r = Node l x r

preOrderNode :: a -> Tree a -> Tree a -> Tree a
preOrderNode x l r = Node l x r

postOrderNode :: Tree a -> Tree a -> a -> Tree a
postOrderNode l r x = Node l x r

-- Traversable instance with in-order traversal
instance Traversable Tree where
    traverse g t = case t of
        Empty      -> pure Empty
        Leaf x     -> Leaf <$> g x
        Node l x r -> inOrderNode <$> traverse g l <*> g x <*> traverse g r

-- Traversable instance with pre-order traversal
instance Traversable Tree where
    traverse g t = case t of
        Empty      -> pure Empty
        Leaf x     -> Leaf <$> g x
        Node l x r -> preOrderNode <$> g x <*> traverse g l <*> traverse g r

-- Traversable instance with post-order traversal
instance Traversable Tree where
    traverse g t = case t of
        Empty      -> pure Empty
        Leaf x     -> Leaf <$> g x
        Node l x r -> postOrderNode <$> traverse g l <*> traverse g r <*> g x

Since the same underlying Tree structure is used in all three cases, it is possible to use newtype wrappers to make all three available at the same time! The user need only wrap the root of the tree in the appropriate newtype for the desired traversal order. Tne associated instance definitions are shown below (see coercion if unfamiliar with the use of coerce in the sample code):

{-# LANGUAGE ScopedTypeVariables, TypeApplications #-}

-- Default in-order traversal

import Data.Coerce (coerce)
import Data.Traversable

data Tree a = Empty | Leaf a | Node (Tree a) a (Tree a)
instance Functor  Tree where fmap    = fmapDefault
instance Foldable Tree where foldMap = foldMapDefault

instance Traversable Tree where
    traverse _ Empty = pure Empty
    traverse g (Leaf a) = Leaf <$> g a
    traverse g (Node l a r) = Node <$> traverse g l <*> g a <*> traverse g r

-- Optional pre-order traversal

newtype PreOrderTree a = PreOrderTree (Tree a)
instance Functor  PreOrderTree where fmap    = fmapDefault
instance Foldable PreOrderTree where foldMap = foldMapDefault

instance Traversable PreOrderTree where
    traverse _ (PreOrderTree Empty)        = pure $ preOrderEmpty
    traverse g (PreOrderTree (Leaf x))     = preOrderLeaf <$> g x
    traverse g (PreOrderTree (Node l x r)) = preOrderNode
        <$> g x
        <*> traverse g (coerce l)
        <*> traverse g (coerce r)

preOrderEmpty :: forall a. PreOrderTree a
preOrderEmpty = coerce (Empty @a)
preOrderLeaf :: forall a. a -> PreOrderTree a
preOrderLeaf = coerce (Leaf @a)
preOrderNode :: a -> PreOrderTree a -> PreOrderTree a -> PreOrderTree a
preOrderNode x l r = coerce (Node (coerce l) x (coerce r))

-- Optional post-order traversal

newtype PostOrderTree a = PostOrderTree (Tree a)
instance Functor  PostOrderTree where fmap    = fmapDefault
instance Foldable PostOrderTree where foldMap = foldMapDefault

instance Traversable PostOrderTree where
    traverse _ (PostOrderTree Empty)        = pure postOrderEmpty
    traverse g (PostOrderTree (Leaf x))     = postOrderLeaf <$> g x
    traverse g (PostOrderTree (Node l x r)) = postOrderNode
        <$> traverse g (coerce l)
        <*> traverse g (coerce r)
        <*> g x

postOrderEmpty :: forall a. PostOrderTree a
postOrderEmpty = coerce (Empty @a)
postOrderLeaf :: forall a. a -> PostOrderTree a
postOrderLeaf = coerce (Leaf @a)
postOrderNode :: PostOrderTree a -> PostOrderTree a -> a -> PostOrderTree a
postOrderNode l r x = coerce (Node (coerce l) x (coerce r))

With the above, given a sample tree:

inOrder :: Tree Int
inOrder = Node (Node (Leaf 10) 3 (Leaf 20)) 5 (Leaf 42)

we have:

import Data.Foldable (toList)
print $ toList inOrder
[10,3,20,5,42]

print $ toList (coerce inOrder :: PreOrderTree Int)
[5,3,10,20,42]

print $ toList (coerce inOrder :: PostOrderTree Int)
[10,20,3,42,5]

You would typically define instances for additional common type classes, such as Eq, Ord, Show, etc.

In order to be able to reason about how a given type of Applicative effects will be sequenced through a general Traversable structure by its traversable and related methods, it is helpful to look more closely at how a general traverse method is implemented. We'll look at how general traversals are constructed primarily with a view to being able to predict their behaviour as a user, even if you're not defining your own Traversable instances.

Traversable structures t a are assembled incrementally from their constituent parts, perhaps by prepending or appending individual elements of type a, or, more generally, by recursively combining smaller composite traversable building blocks that contain multiple such elements.

As in the tree example above, the components being combined are typically pieced together by a suitable constructor, i.e. a function taking two or more arguments that returns a composite value.

The traverse method enriches simple incremental construction with threading of Applicative effects of some function g :: a -> f b.

The basic building blocks we'll use to model the construction of traverse are a hypothetical set of elementary functions, some of which may have direct analogues in specific Traversable structures. For example, the (:) constructor is an analogue for lists of prepend or the more general combine.

empty :: t a               -- build an empty container
singleton :: a -> t a      -- build a one-element container
prepend :: a -> t a -> t a -- extend by prepending a new initial element
append  :: t a -> a -> t a -- extend by appending a new final element
combine :: a1 -> a2 -> ... -> an -> t a -- combine multiple inputs

• An empty structure has no elements of type a, so there's nothing to which g can be applied, but since we need an output of type f (t b), we just use the pure instance of f to wrap an empty of type t b:

  traverse _ (empty :: t a) = pure (empty :: t b)

  With the List monad, empty is [], while with Maybe it is Nothing. With Either e a we have an empty case for each value of e:

  traverse _ (Left e :: Either e a) = pure $ (Left e :: Either e b)

• A singleton structure has just one element of type a, and traverse can take that a, apply g :: a -> f b getting an f b, then fmap singleton over that, getting an f (t b) as required:

  traverse g (singleton a) = fmap singleton $ g a

  Note that if f is List and g returns multiple values the result will be a list of multiple t b singletons!

  Since Maybe and Either are either empty or singletons, we have

  traverse _ Nothing = pure Nothing
  traverse g (Just a) = Just <$> g a

  traverse _ (Left e) = pure (Left e)
  traverse g (Right a) = Right <$> g a

  For List, empty is [] and singleton is (:[]), so we have:

  traverse _ []  = pure []
  traverse g [a] = fmap (:[]) (g a)
                 = (:) <$> (g a) <*> traverse g []
                 = liftA2 (:) (g a) (traverse g [])

• When the structure is built by adding one more element via prepend or append, traversal amounts to:

  traverse g (prepend a t0) = prepend <$> (g a) <*> traverse g t0
                            = liftA2 prepend (g a) (traverse g t0)

  traverse g (append t0 a) = append <$> traverse g t0 <*> g a
                           = liftA2 append (traverse g t0) (g a)

  The origin of the combinatorial product when f is List should now be apparent, when traverse g t0 has n elements and g a has m elements, the non-deterministic Applicative instance of List will produce a result with m * n elements.

• When combining larger building blocks, we again use (<*>) to combine the traversals of the components. With bare elements a mapped to f b via g, and composite traversable sub-structures transformed via traverse g:

  traverse g (combine a1 a2 ... an) =
      combine <$> t1 <*> t2 <*> ... <*> tn
    where
       t1 = g a1          -- if a1 fills a slot of type @a@
          = traverse g a1 -- if a1 is a traversable substructure
       ... ditto for the remaining constructor arguments ...

The above definitions sequence the Applicative effects of f in the expected order while producing results of the expected shape t.

For lists this becomes:

traverse g [] = pure []
traverse g (x:xs) = liftA2 (:) (g a) (traverse g xs)

The actual definition of traverse for lists is an equivalent right fold in order to facilitate list fusion.

traverse g = foldr (\x r -> liftA2 (:) (g x) r) (pure [])

In the sections below we'll examine some advanced choices of Applicative effects that give rise to very different transformations of Traversable structures.

These examples cover the implementations of fmapDefault, foldMapDefault, mapAccumL and mapAccumR functions illustrating the use of Identity, Const and stateful Applicative effects. The ZipList example illustrates the use of a less-well known Applicative instance for lists.

This is optional material, which is not essential to a basic understanding of Traversable structures. If this is your first encounter with Traversable structures, you can come back to these at a later date.

Some of the examples make use of an advanced Haskell feature, namely newtype coercion. This is done for two reasons:

• Use of coerce makes it possible to avoid cluttering the code with functions that wrap and unwrap newtype terms, which at runtime are indistinguishable from the underlying value. Coercion is particularly convenient when one would have to otherwise apply multiple newtype constructors to function arguments, and then peel off multiple layers of same from the function output.
• Use of coerce can produce more efficient code, by reusing the original value, rather than allocating space for a wrapped clone.

If you're not familiar with coerce, don't worry, it is just a shorthand that, e.g., given:

newtype Foo a = MkFoo { getFoo :: a }
newtype Bar a = MkBar { getBar :: a }
newtype Baz a = MkBaz { getBaz :: a }
f :: Baz Int -> Bar (Foo String)

makes it possible to write:

x :: Int -> String
x = coerce f

instead of

x = getFoo . getBar . f . MkBaz

The simplest Applicative functor is Identity, which just wraps and unwraps pure values and function application. This allows us to define fmapDefault:

{-# LANGUAGE ScopedTypeVariables, TypeApplications #-}
import Data.Coercible (coerce)

fmapDefault :: forall t a b. Traversable t => (a -> b) -> t a -> t b
fmapDefault = coerce (traverse @t @Identity @a @b)

The use of coercion avoids the need to explicitly wrap and unwrap terms via Identity and runIdentity.

As noted in Overview, fmapDefault can only be used to define the requisite Functor instance of a Traversable structure when the traverse method is explicitly implemented. An infinite loop would result if in addition traverse were defined in terms of sequenceA and fmap.

Applicative functors that thread a changing state through a computation are an interesting use-case for traverse. The mapAccumL and mapAccumR functions in this module are each defined in terms of such traversals.

We first define a simplified (not a monad transformer) version of State that threads a state s through a chain of computations left to right. Its (<*>) operator passes the input state first to its left argument, and then the resulting state is passed to its right argument, which returns the final state.

newtype StateL s a = StateL { runStateL :: s -> (s, a) }

instance Functor (StateL s) where
    fmap f (StateL kx) = StateL $ \ s ->
        let (s', x) = kx s in (s', f x)

instance Applicative (StateL s) where
    pure a = StateL $ \s -> (s, a)
    (StateL kf) <*> (StateL kx) = StateL $ \ s ->
        let { (s',  f) = kf s
            ; (s'', x) = kx s' } in (s'', f x)
    liftA2 f (StateL kx) (StateL ky) = StateL $ \ s ->
        let { (s',  x) = kx s
            ; (s'', y) = ky s' } in (s'', f x y)

With StateL, we can define mapAccumL as follows:

{-# LANGUAGE ScopedTypeVariables, TypeApplications #-}
mapAccumL :: forall t s a b. Traversable t
          => (s -> a -> (s, b)) -> s -> t a -> (s, t b)
mapAccumL g s ts = coerce (traverse @t @(StateL s) @a @b) (flip g) ts s

The use of coercion avoids the need to explicitly wrap and unwrap newtype terms.

The type of flip g is coercible to a -> StateL b, which makes it suitable for use with traverse. As part of the Applicative construction of StateL (t b) the state updates will thread left-to-right along the sequence of elements of t a.

While mapAccumR has a type signature identical to mapAccumL, it differs in the expected order of evaluation of effects, which must take place right-to-left.

For this we need a variant control structure StateR, which threads the state right-to-left, by passing the input state to its right argument and then using the resulting state as an input to its left argument:

newtype StateR s a = StateR { runStateR :: s -> (s, a) }

instance Functor (StateR s) where
    fmap f (StateR kx) = StateR $ \s ->
        let (s', x) = kx s in (s', f x)

instance Applicative (StateR s) where
    pure a = StateR $ \s -> (s, a)
    (StateR kf) <*> (StateR kx) = StateR $ \ s ->
        let { (s',  x) = kx s
            ; (s'', f) = kf s' } in (s'', f x)
    liftA2 f (StateR kx) (StateR ky) = StateR $ \ s ->
        let { (s',  y) = ky s
            ; (s'', x) = kx s' } in (s'', f x y)

With StateR, we can define mapAccumR as follows:

{-# LANGUAGE ScopedTypeVariables, TypeApplications #-}
mapAccumR :: forall t s a b. Traversable t
          => (s -> a -> (s, b)) -> s -> t a -> (s, t b)
mapAccumR g s0 ts = coerce (traverse @t @(StateR s) @a @b) (flip g) ts s0

The use of coercion avoids the need to explicitly wrap and unwrap newtype terms.

Various stateful traversals can be constructed from mapAccumL and mapAccumR for suitable choices of g, or built directly along similar lines.

The Const Functor enables applications of traverse that summarise the input structure to an output value without constructing any output values of the same type or shape.

As noted above, the Foldable superclass constraint is justified by the fact that it is possible to construct foldMap, foldr, etc., from traverse. The technique used is useful in its own right, and is explored below.

A key feature of folds is that they can reduce the input structure to a summary value. Often neither the input structure nor a mutated clone is needed once the fold is computed, and through list fusion the input may not even have been memory resident in its entirety at the same time.

The traverse method does not at first seem to be a suitable building block for folds, because its return value f (t b) appears to retain mutated copies of the input structure. But the presence of t b in the type signature need not mean that terms of type t b are actually embedded in f (t b). The simplest way to elide the excess terms is by basing the Applicative functor used with traverse on Const.

Not only does Const a b hold just an a value, with the b parameter merely a phantom type, but when m has a Monoid instance, Const m is an Applicative functor:

import Data.Coerce (coerce)
newtype Const a b = Const { getConst :: a } deriving (Eq, Ord, Show) -- etc.
instance Functor (Const m) where fmap = const coerce
instance Monoid m => Applicative (Const m) where
   pure _   = Const mempty
   (<*>)    = coerce (mappend :: m -> m -> m)
   liftA2 _ = coerce (mappend :: m -> m -> m)

The use of coercion avoids the need to explicitly wrap and unwrap newtype terms.

We can therefore define a specialisation of traverse:

{-# LANGUAGE ScopedTypeVariables, TypeApplications #-}
traverseC :: forall t a m. (Monoid m, Traversable t)
          => (a -> Const m ()) -> t a -> Const m (t ())
traverseC = traverse @t @(Const m) @a @()

For which the Applicative construction of traverse leads to:

null ts ==> traverseC g ts = Const mempty

traverseC g (prepend x xs) = Const (g x) <> traverseC g xs

In other words, this makes it possible to define:

{-# LANGUAGE ScopedTypeVariables, TypeApplications #-}
foldMapDefault :: forall t a m. (Monoid m, Traversable t) => (a -> m) -> t a -> m
foldMapDefault = coerce (traverse @t @(Const m) @a @())

Which is sufficient to define a Foldable superclass instance:

The use of coercion avoids the need to explicitly wrap and unwrap newtype terms.

instance Traversable t => Foldable t where foldMap = foldMapDefault

It may however be instructive to also directly define candidate default implementations of foldr and foldl', which take a bit more machinery to construct:

{-# LANGUAGE ScopedTypeVariables, TypeApplications #-}
import Data.Coerce (coerce)
import Data.Functor.Const (Const(..))
import Data.Semigroup (Dual(..), Endo(..))
import GHC.Exts (oneShot)

foldrDefault :: forall t a b. Traversable t
             => (a -> b -> b) -> b -> t a -> b
foldrDefault f z = \t ->
    coerce (traverse @t @(Const (Endo b)) @a @()) f t z

foldlDefault' :: forall t a b. Traversable t => (b -> a -> b) -> b -> t a -> b
foldlDefault' f z = \t ->
    coerce (traverse @t @(Const (Dual (Endo b))) @a @()) f' t z
  where
    f' :: a -> b -> b
    f' a = oneShot $ \ b -> b `seq` f b a

In the above we're using the Endo b Monoid and its Dual to compose a sequence of b -> b accumulator updates in either left-to-right or right-to-left order.

The use of seq in the definition of foldlDefault' ensures strictness in the accumulator.

The use of coercion avoids the need to explicitly wrap and unwrap newtype terms.

The oneShot function gives a hint to the compiler that aids in correct optimisation of lambda terms that fire at most once (for each element a) and so should not try to pre-compute and re-use subexpressions that pay off only on repeated execution. Otherwise, it is just the identity function.

As a warm-up for looking at the ZipList Applicative functor, we'll first look at a simpler analogue. First define a fixed width 2-element Vec2 type, whose Applicative instance combines a pair of functions with a pair of values by applying each function to the corresponding value slot:

data Vec2 a = Vec2 a a
instance Functor Vec2 where
    fmap f (Vec2 a b) = Vec2 (f a) (f b)
instance Applicative Vec2 where
    pure x = Vec2 x x
    liftA2 f (Vec2 a b) (Vec2 p q) = Vec2 (f a p) (f b q)
instance Foldable Vec2 where
    foldr f z (Vec2 a b) = f a (f b z)
    foldMap f (Vec2 a b) = f a <> f b
instance Traversable Vec2 where
    traverse f (Vec2 a b) = Vec2 <$> f a <*> f b

Along with a similar definition for fixed width 3-element vectors:

data Vec3 a = Vec3 a a a
instance Functor Vec3 where
    fmap f (Vec3 x y z) = Vec3 (f x) (f y) (f z)
instance Applicative Vec3 where
    pure x = Vec3 x x x
    liftA2 f (Vec3 p q r) (Vec3 x y z) = Vec3 (f p x) (f q y) (f r z)
instance Foldable Vec3 where
    foldr f z (Vec3 a b c) = f a (f b (f c z))
    foldMap f (Vec3 a b c) = f a <> f b <> f c
instance Traversable Vec3 where
    traverse f (Vec3 a b c) = Vec3 <$> f a <*> f b <*> f c

With the above definitions, sequenceA (same as traverse id) acts as a matrix transpose operation on Vec2 (Vec3 Int) producing a corresponding Vec3 (Vec2 Int):

Let t = Vec2 (Vec3 1 2 3) (Vec3 4 5 6) be our Traversable structure, and g = id :: Vec3 Int -> Vec3 Int be the function used to traverse t. We then have:

traverse g t = Vec2 <$> (Vec3 1 2 3) <*> (Vec3 4 5 6)
             = Vec3 (Vec2 1 4) (Vec2 2 5) (Vec2 3 6)

This construction can be generalised from fixed width vectors to variable length lists via ZipList. This gives a transpose operation that works well for lists of equal length. If some of the lists are longer than others, they're truncated to the longest common length.

We've already looked at the standard Applicative instance of List for which applying m functions f1, f2, ..., fm to n input values a1, a2, ..., an produces m * n outputs:

>>> :set -XTupleSections
>>> [("f1",), ("f2",), ("f3",)] <*> [1,2]
[("f1",1),("f1",2),("f2",1),("f2",2),("f3",1),("f3",2)]

There are however two more common ways to turn lists into Applicative control structures. The first is via Const [a], since lists are monoids under concatenation, and we've already seen that Const m is an Applicative functor when m is a Monoid. The second, is based on zipWith, and is called ZipList:

{-# LANGUAGE GeneralizedNewtypeDeriving #-}
newtype ZipList a = ZipList { getZipList :: [a] }
    deriving (Show, Eq, ..., Functor)

instance Applicative ZipList where
    liftA2 f (ZipList xs) (ZipList ys) = ZipList $ zipWith f xs ys
    pure x = repeat x

The liftA2 definition is clear enough, instead of applying f to each pair (x, y) drawn independently from the xs and ys, only corresponding pairs at each index in the two lists are used.

The definition of pure may look surprising, but it is needed to ensure that the instance is lawful:

liftA2 f (pure x) ys == fmap (f x) ys

Since ys can have any length, we need to provide an infinite supply of x values in pure x in order to have a value to pair with each element y.

When ZipList is the Applicative functor used in the construction of a traversal, a ZipList holding a partially built structure with m elements is combined with a component holding n elements via zipWith, resulting in min m n outputs!

Therefore traverse with g :: a -> ZipList b will produce a ZipList of t b structures whose element count is the minimum length of the ZipLists g a with a ranging over the elements of t. When t is empty, the length is infinite (as expected for a minimum of an empty set).

If the structure t holds values of type ZipList a, we can use the identity function id :: ZipList a -> ZipList a for the first argument of traverse:

traverse (id :: ZipList a -> ZipList a) :: t (ZipList a) -> ZipList (t a)

The number of elements in the output ZipList will be the length of the shortest ZipList element of t. Each output t a will have the same shape as the input t (ZipList a), i.e. will share its number of elements.

If we think of the elements of t (ZipList a) as its rows, and the elements of each individual ZipList as the columns of that row, we see that our traversal implements a transpose operation swapping the rows and columns of t, after first truncating all the rows to the column count of the shortest one.

Since in fact traverse id is just sequenceA the above boils down to a rather concise definition of transpose, with coercion used to implicitly wrap and unwrap the ZipList newtype as needed, giving a function that operates on a list of lists:

>>> :set -XScopedTypeVariables
>>> import Control.Applicative (ZipList(..))
>>> import Data.Coerce (coerce)
>>> 
>>> :{
>>> let
>>> transpose :: forall a. [[a]] -> [[a]]
>>> transpose = coerce (sequenceA :: [ZipList a] -> ZipList [a])
>>> in transpose [[1,2,3],[4..],[7..]]
>>> :}
[[1,4,7],[2,5,8],[3,6,9]]

The use of coercion avoids the need to explicitly wrap and unwrap ZipList terms.

A definition of traverse must satisfy the following laws:

Naturality

t . traverse f = traverse (t . f) for every applicative transformation t

Identity

traverse Identity = Identity

Composition

traverse (Compose . fmap g . f) = Compose . fmap (traverse g) . traverse f

A definition of sequenceA must satisfy the following laws:

Naturality

t . sequenceA = sequenceA . fmap t for every applicative transformation t

Identity

sequenceA . fmap Identity = Identity

Composition

sequenceA . fmap Compose = Compose . fmap sequenceA . sequenceA

where an applicative transformation is a function

t :: (Applicative f, Applicative g) => f a -> g a

preserving the Applicative operations, i.e.

t (pure x) = pure x
t (f <*> x) = t f <*> t x

and the identity functor Identity and composition functors Compose are from Data.Functor.Identity and Data.Functor.Compose.

A result of the naturality law is a purity law for traverse

traverse pure = pure

The superclass instances should satisfy the following:

• In the Functor instance, fmap should be equivalent to traversal with the identity applicative functor (fmapDefault).
• In the Foldable instance, foldMap should be equivalent to traversal with a constant applicative functor (foldMapDefault).

Note: the Functor superclass means that (in GHC) Traversable structures cannot impose any constraints on the element type. A Haskell implementation that supports constrained functors could make it possible to define constrained Traversable structures.

• "The Essence of the Iterator Pattern", by Jeremy Gibbons and Bruno Oliveira, in Mathematically-Structured Functional Programming, 2006, online at http://www.cs.ox.ac.uk/people/jeremy.gibbons/publications/#iterator.
• "Applicative Programming with Effects", by Conor McBride and Ross Paterson, Journal of Functional Programming 18:1 (2008) 1-13, online at http://www.soi.city.ac.uk/~ross/papers/Applicative.html.
• "An Investigation of the Laws of Traversals", by Mauro Jaskelioff and Ondrej Rypacek, in Mathematically-Structured Functional Programming, 2012, online at http://arxiv.org/pdf/1202.2919.