Type classes for mapping, folding, and traversing monomorphic containers

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LTS Haskell 8.3:
Stackage Nightly 2017-03-01:
Latest on Hackage:
MIT licensed by Michael Snoyman, John Wiegley, Greg Weber


Type classes for mapping, folding, and traversing monomorphic and polymorphic containers. Haskell is good at operating over polymorphic containers such as a list [a]. A monomorphic container is one such as Text which has a type Text that does not expose a type variable for the underlying characters.

mono-traversable also adds

  • IsSequence, etc for operating over sequential data types
  • IsSet, IsMap, etc for unifying set and map APIs
  • NonNull for making partial functions (head, tail) total

In addition to this package, the mono-traversable-instances pacakge provides a number of orphan instances.

Using Typeclasses

There are 2 use cases for mono-traversable: application authors and library authors.

Library authors

As a library author, if you want to allow a user to pass in a Text or a String, then you need to expose an API with a mono-traversable typeclass. You should think twice about using mono-traversable though because

  • Using Typeclasses makes type inference more difficult. It is usually better to force the user to give a Text. Another option is to just have multiple APIs.
  • If you are operating on polymorphic structures in which the normal typeclasses suffice, you should just use them from base. On the other hand, even if you are using polymorphic containers you may want to leverage IsSequence or MinLen.

Application authors

As an application author, you should consider using classy-prelude, which leans heavily on mono-traversable.

When writing your own function signatures, you should default to making them concrete: if you are actually using a list, then make your function take a list rather than an IsSequence. This will improve type inference, error messages, and make your code easier to understand. When you decide to use a Vector instead of a list, change the type signature to use a Vector. When you actually need a function to both accept a Vector and a list, it is easy to change the function signature to the more abstract typeclasses that you require.

Standard Typeclasses

in the upcoming GHC 7.10, using Functor, Foldable, and Traversable will become common-place. This means that rather than using,, etc, the map from the prelude will work on all data types that are a Functor. Of course, you can already do this now using fmap.

For a Haskeller, it is important to understand Functor, Applicative, Monad, Foldable, and Monoid: these are encountered in every day code. For mono-traversable, it is most important to understand Foldable.

mono-traversable Typeclasses


Same as Functor, but cannot change the type.

type family   Element mono
type instance Element Text = Char
type instance Element [a] = a

Element is a type family. This tells the compiler to substitute Char for Element Text. We can create this rule for every monomorphic container we want to operate on such as Text And we can also create it for a polymorphic container.

Now lets compare MonoFunctor to the normal Functor.

fmap :: Functor f => (a -> b) -> f a -> f b
omap :: MonoFunctor mono => (Element mono -> Element mono) -> mono -> mono

So there is no type-change from a to b, the contained type must stay the same (Element mono -> Element mono).

Here is the MonoFunctor typeclass definition

class MonoFunctor mono where
    omap :: (Element mono -> Element mono) -> mono -> mono
    default omap :: (Functor f, Element (f a) ~ a, f a ~ mono) => (a -> a) -> f a -> f a
    omap = fmap

And we can write some instances

instance MonoFunctor T.Text where
    omap =

instance MonoFunctor [a]

The list definition was able to default to using fmap so no body was needed.


Same as Foldable, but also operates over monomorphic containers.

MonoFoldable is the heart of the power of mono-traversable (and arguably the package should be named mono-foldable) because anything that can be done with Foldable can be done with MonoFoldable. The reason why is that a monomorphic container can never change its type. So omap is a restricted fmap. However, folding generates a new structure, so we have no such concerns. In the classy-prelude package, map is set to fmap and omap must be used separately. However, foldMap is set to just use the mono-traversable version: ofoldMap

class Foldable t where
  foldMap :: Monoid m => (a -> m) -> t a -> m
  foldr   :: (a -> b -> b) -> b -> t a -> b

class MonoFoldable mono where
  ofoldMap :: Monoid m => (Element mono -> m) -> mono -> m
  ofoldr :: (Element mono -> b -> b) -> b -> mono -> b

There are additional Typeclasses which build on MonoFoldable

class (MonoFoldable mono, Monoid mono) => MonoFoldableMonoid mono where
    oconcatMap :: (Element mono -> mono) -> mono -> mono

class (MonoFoldable mono, Ord (Element mono)) => MonoFoldableOrd mono where
    maximumEx :: mono -> Element mono
    minimumEx :: mono -> Element mono

class MonoPointed mono where
    opoint :: Element mono -> mono

MonoPointed abstracts over the concept of a singleton. For any Applicative, opoint is the same as pure from Applicative. Since mono-traversable did not bother with a MonoApplicative typeclass, we added MonoPointed to still have the functionality of pure.


MonoTraversable is Traversable for monomorphic containers, just as MonoFunctor is Functor for monomorphic containers.

class (Functor t, Foldable t) => Traversable t where
  traverse  :: Applicative f => (a -> f b) -> t a -> f (t b)

class (MonoFunctor mono, MonoFoldable mono) => MonoTraversable mono where
  otraverse :: Applicative f => (Element mono -> f (Element mono)) -> mono -> f mono


  • SetContainer: unifies operations across Set and Map
  • PolyMap: differenceMap and intersectionMap
  • IsSet: unifies operations across different Sets
  • IsMap: unifies operations across different Maps
  • MonoZip: zip operations on MonoFunctors.

Note that because Set and Map are not a Functor (and therefore not MonoFoldable), one must use mapFromList, mapToList, setFromList, and setToList.


IsSequence contains list-like operations.

-- | Sequence Laws:
-- > fromList . otoList = id
-- > fromList (x <> y) = fromList x <> fromList y
-- > otoList (fromList x <> fromList y) = x <> y
class (Monoid seq, MonoTraversable seq, SemiSequence seq, MonoPointed seq) => IsSequence seq where
    fromList :: [Element seq] -> seq
    break :: (Element seq -> Bool) -> seq -> (seq, seq)

The laws state that an IsSequence is a list-like (sequential) structure.

  • an IsSequence is not just something that can be converted to a list (MonoFoldable), but something that can be created from a list.
  • Converting to and from a list does not change the IsSequence, and it doesn't even change the IsSequence if you do the conversions on chunks of the IsSequence.

SemiSequence is required by IsSequence. It is conceptually the same as IsSequence, but contains operations that can also be used on a NonEmpty or a MinLen (which are SemiGroups) because they do not reduce the number of elements in the sequence.

There are some more typeclasess that build on top of IsSequence.

class (IsSequence seq, Eq (Element seq)) => EqSequence seq where
class (EqSequence seq, MonoFoldableOrd seq) => OrdSequence seq where
class (IsSequence t, IsString t, Element t ~ Char) => Textual t where
    words :: t -> [t]
    unwords :: [t] -> t
    lines :: t -> [t]
    unlines :: [t] -> t
    toLower :: t -> t
    toUpper :: t -> t

Textual functions are always safe to use with Unicode (it is possible to misuse other functions that operate on individual characters).


Did you notice minimumEx and maximumEx from above? Ex stands for 'Exception'. An exception will occur if you call minimumEx on an empty list. MinLen is a tool to guarantee that this never occurs, and instead to prove that there are one or more elements in your list.

minimumEx :: MonoFoldable mono => mono -> Element mono

-- | like Data.List, but not partial on a MonoFoldable
minimum :: MonoFoldableOrd mono => MinLen (Succ nat) mono -> Element mono
minimum = minimumEx . unMinLen

newtype MinLen nat mono = MinLen { unMinLen :: mono }
    deriving (Eq, Ord, Read, Show, Data, Typeable, Functor)

-- Type level naturals
data Zero = Zero
data Succ nat = Succ nat

The minimum function exposed from MinLen is very similar to minimumEx, but has a MinLen wrapper that ensures it will never throw an exception. MinLen is a newtype with a phantom type that contains information about the minimum number of elements we know are in the structure. That is done through type-level Peano numbers.

What do we know about the input to minimum? If nat is Zero, then it reduces to MinLen (Succ Zero) mono. Succ means successor, and the successor of 0 is 1, so the data structure has a minimum length of 1.

Lets see this in practice

> minimum []
    Couldn't match expected type ‘MinLen (Succ nat0) mono’
                with actual type ‘[t0]’

> minimum [1,2,3]
-- same error as above

> minimum (toMinList (3 :| [2,1]))
> minimum (3 `mlcons` toMinLenZero [2,1])

Here we used Data.List.NonEmpty combined with toMinList or we just work with a List and prove through the usage of cons that it has more than one element.

Adding instances

If you have a polymorphic data type which is a member of one of the relevant typeclasses (Functor, Foldable, Traversable), it's quite easy to add an instance for MonoFunctor, MonoFoldable or MonoTraversable.

You just have to declare the proper type instance:

{-# LANGUAGE TypeFamilies         #-}

type instance Element (CustomType a) = a

And then, we can use the default implementation to declare instances:

instance MonoFunctor (CustomType a)
instance MonoFoldable (CustomType a)
instance MonoTraversable (CustomType a)

Now you are ready to use `CustomType a` with the functions defined in this package.

Note: if your type is a monomorphic container without the proper typeclasses, then you will have to provide an implementation rather than using the default. However, this should be fairly simple, as can be seen in the code

mono-traversable versus lens Traversal

lens is a library with a lot of functionality covering a variety of patterns. One piece of functionality it exposes is Fold and Traversal which can also be used to deal with monomorphic containers.

You could prefer mono-traversable to using this part of lens because

  • Familiar API - If you know Foldable, you can use MonoFoldable just as easily
  • mono-traversable's typeclass based approach means many methods are included in the class but can be given specialised optimized implementations
  • You don't explicitly pass around the Traversal

The last point is also a point of inflexibility and points to a use case where you could prefer using a lens Traversal. mono-traversable treats ByteString as a sequence of bytes. If you want to treat it as both bytes and characters, mono-traversable would require a newtype wrapper around ByteString, whereas a lens Traversal would use a different traversal function.

mono-traversable is only an alternative for Fold and Traversal, not for Lens, Prism, Iso, Getter, Setter, Review, or Equality.

Build Status


  • Make 'olength' for Set and Map O(1) #125

  • Support for GHC 8.2

  • Fix typo in rewrite rule


  • Add replaceElem and replaceSeq #107

  • Add missing export #101


  • Implement the cleanups described in #95
    • Split out Data.MinLen to minlen package, and have Data.NonNull stand on its own
    • Remove Data.ByteVector
    • Split out extra typeclass instances to mono-traversable-instances
  • Remove the Eq and Ord specific classes, and instead use rewrite rules
  • Provide the Data.MonoTraversable.Unprefixed module
  • Generalize unwords and unlines #87
  • Add tailMay and initMay #89


  • Add delete and deleteBy methods to EqSequence #94

  • Remove unneeded INLINEs #90


  • Allow comonad-5 #86

  • Instance for Data.Sequence.Seq is incorrect. #83


  • Remove Functor instance for MinLen #82


  • Added intercalate, splitWhen, splitElem, and splitSeq #80

  • Tweak test suite for 32-bit systems #78


  • MonoComonad


  • Fill in missing Mono* instances #72

  • Documentation improvements


  • Better fixity for mlcons #56

README updates


A new MonoFoldableEq class that takes elem and notElem from EqSequence. EqSequence now inherits from MonoFoldableEq.

For most users that do not define instances this should not be a breaking change. However, any instance of EqSequence now needs to definie MonoFoldableEq.


  • Work on better polymorphic containers
    • Rename mapKeysWith to omapKeysWith
    • Add new class BiPolyMap
    • Add keys to IsSet
    • New class HasKeysSet
  • Added index, indexEx and unsafeIndex.
  • Added sortOn
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