Difference between revisions of "Typeclassopedia"
m (→Foldable: update doc link) 

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The <code>Foldable</code> class, defined in the <code>Data.Foldable</code> 
The <code>Foldable</code> class, defined in the <code>Data.Foldable</code> 

−  module ([ 
+  module ([https://hackage.haskell.org/package/base4.8.1.0/docs/DataFoldable.html haddock]), abstracts over containers which can be 
“folded” into a summary value. This allows such folding operations 
“folded” into a summary value. This allows such folding operations 

to be written in a containeragnostic way. 
to be written in a containeragnostic way. 
Revision as of 09:21, 3 December 2015
By Brent Yorgey, byorgey@cis.upenn.edu
Originally published 12 March 2009 in issue 13 of the Monad.Reader. Ported to the Haskell wiki in November 2011 by Geheimdienst.
This is now the official version of the Typeclassopedia and supersedes the version published in the Monad.Reader. Please help update and extend it by editing it yourself or by leaving comments, suggestions, and questions on the talk page.
Contents
Abstract
The standard Haskell libraries feature a number of type classes with algebraic or categorytheoretic underpinnings. Becoming a fluent Haskell hacker requires intimate familiarity with them all, yet acquiring this familiarity often involves combing through a mountain of tutorials, blog posts, mailing list archives, and IRC logs.
The goal of this document is to serve as a starting point for the student of Haskell wishing to gain a firm grasp of its standard type classes. The essentials of each type class are introduced, with examples, commentary, and extensive references for further reading.
Introduction
Have you ever had any of the following thoughts?
 What the heck is a monoid, and how is it different from a monad?
 I finally figured out how to use Parsec with donotation, and someone told me I should use something called
Applicative
instead. Um, what?
 Someone in the #haskell IRC channel used
(***)
, and when I asked Lambdabot to tell me its type, it printed out scary gobbledygook that didn’t even fit on one line! Then someone usedfmap fmap fmap
and my brain exploded.
 When I asked how to do something I thought was really complicated, people started typing things like
zip.ap fmap.(id &&& wtf)
and the scary thing is that they worked! Anyway, I think those people must actually be robots because there’s no way anyone could come up with that in two seconds off the top of their head.
If you have, look no further! You, too, can write and understand concise, elegant, idiomatic Haskell code with the best of them.
There are two keys to an expert Haskell hacker’s wisdom:
 Understand the types.
 Gain a deep intuition for each type class and its relationship to other type classes, backed up by familiarity with many examples.
It’s impossible to overstate the importance of the first; the patient student of type signatures will uncover many profound secrets. Conversely, anyone ignorant of the types in their code is doomed to eternal uncertainty. “Hmm, it doesn’t compile ... maybe I’ll stick in an
fmap
here ... nope, let’s see ... maybe I need another (.)
somewhere? ... um ...”
The second key—gaining deep intuition, backed by examples—is also important, but much more difficult to attain. A primary goal of this document is to set you on the road to gaining such intuition. However—
 There is no royal road to Haskell. —Euclid
This document can only be a starting point, since good intuition comes from hard work, not from learning the right metaphor. Anyone who reads and understands all of it will still have an arduous journey ahead—but sometimes a good starting point makes a big difference.
It should be noted that this is not a Haskell tutorial; it is assumed that the reader is already familiar with the basics of Haskell, including the standard Prelude
, the type system, data types, and type classes.
The type classes we will be discussing and their interrelationships:
∗ Semigroup
can be found in the semigroups
package, Apply
in the semigroupoids
package, and Comonad
in the comonad
package.
 Solid arrows point from the general to the specific; that is, if there is an arrow from
Foo
toBar
it means that everyBar
is (or should be, or can be made into) aFoo
.  Dotted arrows indicate some other sort of relationship.

Monad
andArrowApply
are equivalent. 
Semigroup
,Apply
andComonad
are greyed out since they are not actually (yet?) in the standard Haskell libraries ∗.
One more note before we begin. The original spelling of “type class” is with two words, as evidenced by, for example, the Haskell 2010 Language Report, early papers on type classes like Type classes in Haskell and Type classes: exploring the design space, and Hudak et al.’s history of Haskell. However, as often happens with twoword phrases that see a lot of use, it has started to show up as one word (“typeclass”) or, rarely, hyphenated (“typeclass”). When wearing my prescriptivist hat, I prefer “type class”, but realize (after changing into my descriptivist hat) that there's probably not much I can do about it.
We now begin with the simplest type class of all: Functor
.
Functor
The Functor
class (haddock) is the most basic and ubiquitous type class in the Haskell libraries. A simple intuition is that a Functor
represents a “container” of some sort, along with the ability to apply a function uniformly to every element in the container. For example, a list is a container of elements, and we can apply a function to every element of a list, using map
. As another example, a binary tree is also a container of elements, and it’s not hard to come up with a way to recursively apply a function to every element in a tree.
Another intuition is that a Functor
represents some sort of “computational context”. This intuition is generally more useful, but is more difficult to explain, precisely because it is so general. Some examples later should help to clarify the Functor
ascontext point of view.
In the end, however, a Functor
is simply what it is defined to be; doubtless there are many examples of Functor
instances that don’t exactly fit either of the above intuitions. The wise student will focus their attention on definitions and examples, without leaning too heavily on any particular metaphor. Intuition will come, in time, on its own.
Definition
Here is the type class declaration for Functor
:
class Functor f where
fmap :: (a > b) > f a > f b
Functor
is exported by the Prelude
, so no special imports are needed to use it.
First, the f a
and f b
in the type signature for fmap
tell us that f
isn’t just a type; it is a type constructor which takes another type as a parameter. (A more precise way to say this is that the kind of f
must be * > *
.) For example, Maybe
is such a type constructor: Maybe
is not a type in and of itself, but requires another type as a parameter, like Maybe Integer
. So it would not make sense to say instance Functor Integer
, but it could make sense to say instance Functor Maybe
.
Now look at the type of fmap
: it takes any function from a
to b
, and a value of type f a
, and outputs a value of type f b
. From the container point of view, the intention is that fmap
applies a function to each element of a container, without altering the structure of the container. From the context point of view, the intention is that fmap
applies a function to a value without altering its context. Let’s look at a few specific examples.
Instances
∗ Recall that []
has two meanings in Haskell: it can either stand for the empty list, or, as here, it can represent the list type constructor (pronounced “listof”). In other words, the type [a]
(listofa
) can also be written [] a
.
∗ You might ask why we need a separate map
function. Why not just do away with the current listonly map
function, and rename fmap
to map
instead? Well, that’s a good question. The usual argument is that someone just learning Haskell, when using map
incorrectly, would much rather see an error about lists than about Functor
s.
As noted before, the list constructor []
is a functor ∗; we can use the standard list function map
to apply a function to each element of a list ∗. The Maybe
type constructor is also a functor, representing a container which might hold a single element. The function fmap g
has no effect on Nothing
(there are no elements to which g
can be applied), and simply applies g
to the single element inside a Just
. Alternatively, under the context interpretation, the list functor represents a context of nondeterministic choice; that is, a list can be thought of as representing a single value which is nondeterministically chosen from among several possibilities (the elements of the list). Likewise, the Maybe
functor represents a context with possible failure. These instances are:
instance Functor [] where
fmap _ [] = []
fmap g (x:xs) = g x : fmap g xs
 or we could just say fmap = map
instance Functor Maybe where
fmap _ Nothing = Nothing
fmap g (Just a) = Just (g a)
As an aside, in idiomatic Haskell code you will often see the letter f
used to stand for both an arbitrary Functor
and an arbitrary function. In this document, f
represents only Functor
s, and g
or h
always represent functions, but you should be aware of the potential confusion. In practice, what f
stands for should always be clear from the context, by noting whether it is part of a type or part of the code.
There are other Functor
instances in the standard libraries; below are a few. Note that some of these instances are not exported by the Prelude
; to access them, you can import Control.Monad.Instances
.

Either e
is an instance ofFunctor
;Either e a
represents a container which can contain either a value of typea
, or a value of typee
(often representing some sort of error condition). It is similar toMaybe
in that it represents possible failure, but it can carry some extra information about the failure as well.

((,) e)
represents a container which holds an “annotation” of typee
along with the actual value it holds. It might be clearer to write it as(e,)
, by analogy with an operator section like(1+)
, but that syntax is not allowed in types (although it is allowed in expressions with theTupleSections
extension enabled). However, you can certainly think of it as(e,)
.

((>) e)
(which can be thought of as(e >)
; see above), the type of functions which take a value of typee
as a parameter, is aFunctor
. As a container,(e > a)
represents a (possibly infinite) set of values ofa
, indexed by values ofe
. Alternatively, and more usefully,((>) e)
can be thought of as a context in which a value of typee
is available to be consulted in a readonly fashion. This is also why((>) e)
is sometimes referred to as the reader monad; more on this later.

IO
is aFunctor
; a value of typeIO a
represents a computation producing a value of typea
which may have I/O effects. Ifm
computes the valuex
while producing some I/O effects, thenfmap g m
will compute the valueg x
while producing the same I/O effects.
 Many standard types from the containers library (such as
Tree
,Map
, andSequence
) are instances ofFunctor
. A notable exception isSet
, which cannot be made aFunctor
in Haskell (although it is certainly a mathematical functor) since it requires anOrd
constraint on its elements;fmap
must be applicable to any typesa
andb
. However,Set
(and other similarly restricted data types) can be made an instance of a suitable generalization ofFunctor
, either by makinga
andb
arguments to theFunctor
type class themselves, or by adding an associated constraint.
Exercises 


Laws
As far as the Haskell language itself is concerned, the only requirement to be a Functor
is an implementation of fmap
with the proper type. Any sensible Functor
instance, however, will also satisfy the functor laws, which are part of the definition of a mathematical functor. There are two:
fmap id = id
fmap (g . h) = (fmap g) . (fmap h)
∗ Technically, these laws make f
and fmap
together an endofunctor on Hask, the category of Haskell types (ignoring ⊥, which is a party pooper). See Wikibook: Category theory.
Together, these laws ensure that fmap g
does not change the structure of a container, only the elements. Equivalently, and more simply, they ensure that fmap g
changes a value without altering its context ∗.
The first law says that mapping the identity function over every item in a container has no effect. The second says that mapping a composition of two functions over every item in a container is the same as first mapping one function, and then mapping the other.
As an example, the following code is a “valid” instance of Functor
(it typechecks), but it violates the functor laws. Do you see why?
 Evil Functor instance
instance Functor [] where
fmap _ [] = []
fmap g (x:xs) = g x : g x : fmap g xs
Any Haskeller worth their salt would reject this code as a gruesome abomination.
Unlike some other type classes we will encounter, a given type has at most one valid instance of Functor
. This can be proven via the free theorem for the type of fmap
. In fact, GHC can automatically derive Functor
instances for many data types.
∗ Actually, if seq
/undefined
are considered, it is possible to have an implementation which satisfies the first law but not the second. The rest of the comments in this section should considered in a context where seq
and undefined
are excluded.
A similar argument also shows that any Functor
instance satisfying the first law (fmap id = id
) will automatically satisfy the second law as well. Practically, this means that only the first law needs to be checked (usually by a very straightforward induction) to ensure that a Functor
instance is valid.∗
Exercises 


Intuition
There are two fundamental ways to think about fmap
. The first has already been mentioned: it takes two parameters, a function and a container, and applies the function “inside” the container, producing a new container. Alternately, we can think of fmap
as applying a function to a value in a context (without altering the context).
Just like all other Haskell functions of “more than one parameter”, however, fmap
is actually curried: it does not really take two parameters, but takes a single parameter and returns a function. For emphasis, we can write fmap
’s type with extra parentheses: fmap :: (a > b) > (f a > f b)
. Written in this form, it is apparent that fmap
transforms a “normal” function (g :: a > b
) into one which operates over containers/contexts (fmap g :: f a > f b
). This transformation is often referred to as a lift; fmap
“lifts” a function from the “normal world” into the “f
world”.
Further reading
A good starting point for reading about the category theory behind the concept of a functor is the excellent Haskell wikibook page on category theory.
Applicative
A somewhat newer addition to the pantheon of standard Haskell type classes, applicative functors represent an abstraction lying in between Functor
and Monad
in expressivity, first described by McBride and Paterson. The title of their classic paper, Applicative Programming with Effects, gives a hint at the intended intuition behind the Applicative
type class. It encapsulates certain sorts of “effectful” computations in a functionally pure way, and encourages an “applicative” programming style. Exactly what these things mean will be seen later.
Definition
Recall that Functor
allows us to lift a “normal” function to a function on computational contexts. But fmap
doesn’t allow us to apply a function which is itself in a context to a value in a context. Applicative
gives us just such a tool, (<*>)
(variously pronounced as "apply", "app", or "splat"). It also provides a method, pure
, for embedding values in a default, “effect free” context. Here is the type class declaration for Applicative
, as defined in Control.Applicative
:
class Functor f => Applicative f where
pure :: a > f a
infixl 4 <*>
(<*>) :: f (a > b) > f a > f b
Note that every Applicative
must also be a Functor
. In fact, as we will see, fmap
can be implemented using the Applicative
methods, so every Applicative
is a functor whether we like it or not; the Functor
constraint forces us to be honest.
∗ Recall that ($)
is just function application: f $ x = f x
.
As always, it’s crucial to understand the type signatures. First, consider (<*>)
: the best way of thinking about it comes from noting that the type of (<*>)
is similar to the type of ($)
∗, but with everything enclosed in an f
. In other words, (<*>)
is just function application within a computational context. The type of (<*>)
is also very similar to the type of fmap
; the only difference is that the first parameter is f (a > b)
, a function in a context, instead of a “normal” function (a > b)
.
pure
takes a value of any type a
, and returns a context/container of type f a
. The intention is that pure
creates some sort of “default” container or “effect free” context. In fact, the behavior of pure
is quite constrained by the laws it should satisfy in conjunction with (<*>)
. Usually, for a given implementation of (<*>)
there is only one possible implementation of pure
.
(Note that previous versions of the Typeclassopedia explained pure
in terms of a type class Pointed
, which can still be found in the pointed
package. However, the current consensus is that Pointed
is not very useful after all. For a more detailed explanation, see Why not Pointed?)
Laws
∗ See haddock for Applicative and Applicative programming with effects
Traditionally, there are four laws that Applicative
instances should satisfy ∗. In some sense, they are all concerned with making sure that pure
deserves its name:
 The identity law:
pure id <*> v = v
 Homomorphism:Intuitively, applying a noneffectful function to a noneffectful argument in an effectful context is the same as just applying the function to the argument and then injecting the result into the context with
pure f <*> pure x = pure (f x)
pure
.  Interchange:Intuitively, this says that when evaluating the application of an effectful function to a pure argument, the order in which we evaluate the function and its argument doesn't matter.
u <*> pure y = pure ($ y) <*> u
 Composition:This one is the trickiest law to gain intuition for. In some sense it is expressing a sort of associativity property of
u <*> (v <*> w) = pure (.) <*> u <*> v <*> w
(<*>)
. The reader may wish to simply convince themselves that this law is typecorrect.
Considered as lefttoright rewrite rules, the homomorphism, interchange, and composition laws actually constitute an algorithm for transforming any expression using pure
and (<*>)
into a canonical form with only a single use of pure
at the very beginning and only leftnested occurrences of (<*>)
. Composition allows reassociating (<*>)
; interchange allows moving occurrences of pure
leftwards; and homomorphism allows collapsing multiple adjacent occurrences of pure
into one.
There is also a law specifying how Applicative
should relate to Functor
:
fmap g x = pure g <*> x
It says that mapping a pure function g
over a context x
is the same as first injecting g
into a context with pure
, and then applying it to x
with (<*>)
. In other words, we can decompose fmap
into two more atomic operations: injection into a context, and application within a context. The Control.Applicative
module also defines (<$>)
as a synonym for fmap
, so the above law can also be expressed as:
g <$> x = pure g <*> x
.
Exercises 


Instances
Most of the standard types which are instances of Functor
are also instances of Applicative
.
Maybe
can easily be made an instance of Applicative
; writing such an instance is left as an exercise for the reader.
The list type constructor []
can actually be made an instance of Applicative
in two ways; essentially, it comes down to whether we want to think of lists as ordered collections of elements, or as contexts representing multiple results of a nondeterministic computation (see Wadler’s How to replace failure by a list of successes).
Let’s first consider the collection point of view. Since there can only be one instance of a given type class for any particular type, one or both of the list instances of Applicative
need to be defined for a newtype
wrapper; as it happens, the nondeterministic computation instance is the default, and the collection instance is defined in terms of a newtype
called ZipList
. This instance is:
newtype ZipList a = ZipList { getZipList :: [a] }
instance Applicative ZipList where
pure = undefined  exercise
(ZipList gs) <*> (ZipList xs) = ZipList (zipWith ($) gs xs)
To apply a list of functions to a list of inputs with (<*>)
, we just match up the functions and inputs elementwise, and produce a list of the resulting outputs. In other words, we “zip” the lists together with function application, ($)
; hence the name ZipList
.
The other Applicative
instance for lists, based on the nondeterministic computation point of view, is:
instance Applicative [] where
pure x = [x]
gs <*> xs = [ g x  g < gs, x < xs ]
Instead of applying functions to inputs pairwise, we apply each function to all the inputs in turn, and collect all the results in a list.
Now we can write nondeterministic computations in a natural style. To add the numbers 3
and 4
deterministically, we can of course write (+) 3 4
. But suppose instead of 3
we have a nondeterministic computation that might result in 2
, 3
, or 4
; then we can write
pure (+) <*> [2,3,4] <*> pure 4
or, more idiomatically,
(+) <$> [2,3,4] <*> pure 4.
There are several other Applicative
instances as well:

IO
is an instance ofApplicative
, and behaves exactly as you would think: to executem1 <*> m2
, firstm1
is executed, resulting in a functionf
, thenm2
is executed, resulting in a valuex
, and finally the valuef x
is returned as the result of executingm1 <*> m2
.

((,) a)
is anApplicative
, as long asa
is an instance ofMonoid
(section Monoid). Thea
values are accumulated in parallel with the computation.
 The
Applicative
module defines theConst
type constructor; a value of typeConst a b
simply contains ana
. This is an instance ofApplicative
for anyMonoid a
; this instance becomes especially useful in conjunction with things likeFoldable
(section Foldable).
 The
WrappedMonad
andWrappedArrow
newtypes make any instances ofMonad
(section Monad) orArrow
(section Arrow) respectively into instances ofApplicative
; as we will see when we study those type classes, both are strictly more expressive thanApplicative
, in the sense that theApplicative
methods can be implemented in terms of their methods.
Exercises 


Intuition
McBride and Paterson’s paper introduces the notation to denote function application in a computational context. If each has type for some applicative functor , and has type , then the entire expression has type . You can think of this as applying a function to multiple “effectful” arguments. In this sense, the double bracket notation is a generalization of fmap
, which allows us to apply a function to a single argument in a context.
Why do we need Applicative
to implement this generalization of fmap
? Suppose we use fmap
to apply g
to the first parameter x1
. Then we get something of type f (t2 > ... t)
, but now we are stuck: we can’t apply this functioninacontext to the next argument with fmap
. However, this is precisely what (<*>)
allows us to do.
This suggests the proper translation of the idealized notation into Haskell, namely
g <$> x1 <*> x2 <*> ... <*> xn,
recalling that Control.Applicative
defines (<$>)
as convenient infix shorthand for fmap
. This is what is meant by an “applicative style”—effectful computations can still be described in terms of function application; the only difference is that we have to use the special operator (<*>)
for application instead of simple juxtaposition.
Note that pure
allows embedding “noneffectful” arguments in the middle of an idiomatic application, like
g <$> x1 <*> pure x2 <*> x3
which has type f d
, given
g :: a > b > c > d
x1 :: f a
x2 :: b
x3 :: f c
The double brackets are commonly known as “idiom brackets”, because they allow writing “idiomatic” function application, that is, function application that looks normal but has some special, nonstandard meaning (determined by the particular instance of Applicative
being used). Idiom brackets are not supported by GHC, but they are supported by the Strathclyde Haskell Enhancement, a preprocessor which (among many other things) translates idiom brackets into standard uses of (<$>)
and (<*>)
. This can result in much more readable code when making heavy use of Applicative
.
Alternative formulation
An alternative, equivalent formulation of Applicative
is given by
class Functor f => Monoidal f where
unit :: f ()
(**) :: f a > f b > f (a,b)
∗ In categorytheory speak, we say f
is a lax monoidal functor because there aren't necessarily functions in the other direction, like f (a, b) > (f a, f b)
.
Intuitively, this states that a monoidal functor∗ is one which has some sort of "default shape" and which supports some sort of "combining" operation. pure
and (<*>)
are equivalent in power to unit
and (**)
(see the Exercises below). More technically, the idea is that f
preserves the "monoidal structure" given by the pairing constructor (,)
and unit type ()
. This can be seen even more clearly if we rewrite the types of unit
and (**)
as
unit' :: () > f ()
(**') :: (f a, f b) > f (a, b)
Furthermore, to deserve the name "monoidal" (see the section on Monoids), instances of Monoidal
ought to satisfy the following laws, which seem much more straightforward than the traditional Applicative
laws:
∗ In this and the following laws, ≅
refers to isomorphism rather than equality. In particular we consider (x,()) ≅ x ≅ ((),x)
and ((x,y),z) ≅ (x,(y,z))
.
 Left identity∗:
unit ** v ≅ v
 Right identity:
u ** unit ≅ u
 Associativity:
u ** (v ** w) ≅ (u ** v) ** w
These turn out to be equivalent to the usual Applicative
laws. In a category theory setting, one would also require a naturality law:
∗ Here g *** h = \(x,y) > (g x, h y)
. See Arrows.
 Naturality:
fmap (g *** h) (u ** v) = fmap g u ** fmap h v
but in the context of Haskell, this is a free theorem.
Much of this section was taken from a blog post by Edward Z. Yang; see his actual post for a bit more information.
Exercises 


Further reading
There are many other useful combinators in the standard libraries implemented in terms of pure
and (<*>)
: for example, (*>)
, (<*)
, (<**>)
, (<$)
, and so on (see haddock for Applicative). Judicious use of such secondary combinators can often make code using Applicative
s much easier to read.
McBride and Paterson’s original paper is a treasuretrove of information and examples, as well as some perspectives on the connection between Applicative
and category theory. Beginners will find it difficult to make it through the entire paper, but it is extremely wellmotivated—even beginners will be able to glean something from reading as far as they are able.
∗ Introduced by an earlier paper that was since superseded by Pushpull functional reactive programming.
Conal Elliott has been one of the biggest proponents of Applicative
. For example, the Pan library for functional images and the reactive library for functional reactive programming (FRP) ∗ make key use of it; his blog also contains many examples of Applicative
in action. Building on the work of McBride and Paterson, Elliott also built the TypeCompose library, which embodies the observation (among others) that Applicative
types are closed under composition; therefore, Applicative
instances can often be automatically derived for complex types built out of simpler ones.
Although the Parsec parsing library (paper) was originally designed for use as a monad, in its most common use cases an Applicative
instance can be used to great effect; Bryan O’Sullivan’s blog post is a good starting point. If the extra power provided by Monad
isn’t needed, it’s usually a good idea to use Applicative
instead.
A couple other nice examples of Applicative
in action include the ConfigFile and HSQL libraries and the formlets library.
Gershom Bazerman's post contains many insights into applicatives.
Monad
It’s a safe bet that if you’re reading this, you’ve heard of monads—although it’s quite possible you’ve never heard of Applicative
before, or Arrow
, or even Monoid
. Why are monads such a big deal in Haskell? There are several reasons.
 Haskell does, in fact, single out monads for special attention by making them the framework in which to construct I/O operations.
 Haskell also singles out monads for special attention by providing a special syntactic sugar for monadic expressions: the
do
notation. 
Monad
has been around longer than other abstract models of computation such asApplicative
orArrow
.  The more monad tutorials there are, the harder people think monads must be, and the more new monad tutorials are written by people who think they finally “get” monads (the monad tutorial fallacy).
I will let you judge for yourself whether these are good reasons.
In the end, despite all the hoopla, Monad
is just another type class. Let’s take a look at its definition.
Definition
The type class declaration for Monad
is:
class Monad m where
return :: a > m a
(>>=) :: m a > (a > m b) > m b
(>>) :: m a > m b > m b
m >> n = m >>= \_ > n
fail :: String > m a
The Monad
type class is exported by the Prelude
, along with a few standard instances. However, many utility functions are found in Control.Monad
, and there are also several instances (such as ((>) e)
) defined in Control.Monad.Instances
.
∗ However, as of GHC 7.10 this will be fixed!
Let’s examine the methods in the Monad
class one by one. The type of return
should look familiar; it’s the same as pure
. Indeed, return
is pure
, but with an unfortunate name. (Unfortunate, since someone coming from an imperative programming background might think that return
is like the C or Java keyword of the same name, when in fact the similarities are minimal.) From a mathematical point of view, every monad is an applicative functor, but for historical reasons, the Monad
type class declaration unfortunately does not require this.∗
We can see that (>>)
is a specialized version of (>>=)
, with a default implementation given. It is only included in the type class declaration so that specific instances of Monad
can override the default implementation of (>>)
with a more efficient one, if desired. Also, note that although _ >> n = n
would be a typecorrect implementation of (>>)
, it would not correspond to the intended semantics: the intention is that m >> n
ignores the result of m
, but not its effects.
The fail
function is an awful hack that has no place in the Monad
class; more on this later.
The only really interesting thing to look at—and what makes Monad
strictly more powerful than Applicative
—is (>>=)
, which is often called bind. An alternative definition of Monad
could look like:
class Applicative m => Monad' m where
(>>=) :: m a > (a > m b) > m b
We could spend a while talking about the intuition behind (>>=)
—and we will. But first, let’s look at some examples.
Instances
Even if you don’t understand the intuition behind the Monad
class, you can still create instances of it by just seeing where the types lead you. You may be surprised to find that this actually gets you a long way towards understanding the intuition; at the very least, it will give you some concrete examples to play with as you read more about the Monad
class in general. The first few examples are from the standard Prelude
; the remaining examples are from the transformers
package.
 The simplest possible instance of
Monad
isIdentity
, which is described in Dan Piponi’s highly recommended blog post on The Trivial Monad. Despite being “trivial”, it is a great introduction to theMonad
type class, and contains some good exercises to get your brain working.  The next simplest instance of
Monad
isMaybe
. We already know how to writereturn
/pure
forMaybe
. So how do we write(>>=)
? Well, let’s think about its type. Specializing forMaybe
, we have(>>=) :: Maybe a > (a > Maybe b) > Maybe b.
If the first argument to
(>>=)
isJust x
, then we have something of typea
(namely,x
), to which we can apply the second argument—resulting in aMaybe b
, which is exactly what we wanted. What if the first argument to(>>=)
isNothing
? In that case, we don’t have anything to which we can apply thea > Maybe b
function, so there’s only one thing we can do: yieldNothing
. This instance is:instance Monad Maybe where return = Just (Just x) >>= g = g x Nothing >>= _ = Nothing
We can already get a bit of intuition as to what is going on here: if we build up a computation by chaining together a bunch of functions with
(>>=)
, as soon as any one of them fails, the entire computation will fail (becauseNothing >>= f
isNothing
, no matter whatf
is). The entire computation succeeds only if all the constituent functions individually succeed. So theMaybe
monad models computations which may fail.  The
Monad
instance for the list constructor[]
is similar to itsApplicative
instance; see the exercise below.  Of course, the
IO
constructor is famously aMonad
, but its implementation is somewhat magical, and may in fact differ from compiler to compiler. It is worth emphasizing that theIO
monad is the only monad which is magical. It allows us to build up, in an entirely pure way, values representing possibly effectful computations. The special valuemain
, of typeIO ()
, is taken by the runtime and actually executed, producing actual effects. Every other monad is functionally pure, and requires no special compiler support. We often speak of monadic values as “effectful computations”, but this is because some monads allow us to write code as if it has side effects, when in fact the monad is hiding the plumbing which allows these apparent side effects to be implemented in a functionally pure way.  As mentioned earlier,
((>) e)
is known as the reader monad, since it describes computations in which a value of typee
is available as a readonly environment. TheControl.Monad.Reader
module provides theReader e a
type, which is just a convenientnewtype
wrapper around(e > a)
, along with an appropriateMonad
instance and someReader
specific utility functions such asask
(retrieve the environment),asks
(retrieve a function of the environment), andlocal
(run a subcomputation under a different environment).  The
Control.Monad.Writer
module provides theWriter
monad, which allows information to be collected as a computation progresses.Writer w a
is isomorphic to(a,w)
, where the output valuea
is carried along with an annotation or “log” of typew
, which must be an instance ofMonoid
(see section Monoid); the special functiontell
performs logging.  The
Control.Monad.State
module provides theState s a
type, anewtype
wrapper arounds > (a,s)
. Something of typeState s a
represents a stateful computation which produces ana
but can access and modify the state of types
along the way. The module also providesState
specific utility functions such asget
(read the current state),gets
(read a function of the current state),put
(overwrite the state), andmodify
(apply a function to the state).  The
Control.Monad.Cont
module provides theCont
monad, which represents computations in continuationpassing style. It can be used to suspend and resume computations, and to implement nonlocal transfers of control, coroutines, other complex control structures—all in a functionally pure way.Cont
has been called the “mother of all monads” because of its universal properties.
Exercises 


Intuition
Let’s look more closely at the type of (>>=)
. The basic intuition is that it combines two computations into one larger computation. The first argument, m a
, is the first computation. However, it would be boring if the second argument were just an m b
; then there would be no way for the computations to interact with one another (actually, this is exactly the situation with Applicative
). So, the second argument to (>>=)
has type a > m b
: a function of this type, given a result of the first computation, can produce a second computation to be run. In other words, x >>= k
is a computation which runs x
, and then uses the result(s) of x
to decide what computation to run second, using the output of the second computation as the result of the entire computation.
∗ Actually, because Haskell allows general recursion, this is a lie: using a Haskell parsing library one can recursively construct infinite grammars, and hence Applicative
(together with Alternative
) is enough to parse any contextsensitive language with a finite alphabet. See Parsing contextsensitive languages with Applicative.
Intuitively, it is this ability to use the output from previous computations to decide what computations to run next that makes Monad
more powerful than Applicative
. The structure of an Applicative
computation is fixed, whereas the structure of a Monad
computation can change based on intermediate results. This also means that parsers built using an Applicative
interface can only parse contextfree languages; in order to parse contextsensitive languages a Monad
interface is needed.∗
To see the increased power of Monad
from a different point of view, let’s see what happens if we try to implement (>>=)
in terms of fmap
, pure
, and (<*>)
. We are given a value x
of type m a
, and a function k
of type a > m b
, so the only thing we can do is apply k
to x
. We can’t apply it directly, of course; we have to use fmap
to lift it over the m
. But what is the type of fmap k
? Well, it’s m a > m (m b)
. So after we apply it to x
, we are left with something of type m (m b)
—but now we are stuck; what we really want is an m b
, but there’s no way to get there from here. We can add m
’s using pure
, but we have no way to collapse multiple m
’s into one.
∗ You might hear some people claim that that the definition in terms of return
, fmap
, and join
is the “math definition” and the definition in terms of return
and (>>=)
is something specific to Haskell. In fact, both definitions were known in the mathematics community long before Haskell picked up monads.
This ability to collapse multiple m
’s is exactly the ability provided by the function join :: m (m a) > m a
, and it should come as no surprise that an alternative definition of Monad
can be given in terms of join
:
class Applicative m => Monad'' m where
join :: m (m a) > m a
In fact, the canonical definition of monads in category theory is in terms of return
, fmap
, and join
(often called , , and in the mathematical literature). Haskell uses an alternative formulation with (>>=)
instead of join
since it is more convenient to use ∗. However, sometimes it can be easier to think about Monad
instances in terms of join
, since it is a more “atomic” operation. (For example, join
for the list monad is just concat
.)
Exercises 


Utility functions
The Control.Monad
module provides a large number of convenient utility functions, all of which can be implemented in terms of the basic Monad
operations (return
and (>>=)
in particular). We have already seen one of them, namely, join
. We also mention some other noteworthy ones here; implementing these utility functions oneself is a good exercise. For a more detailed guide to these functions, with commentary and example code, see HenkJan van Tuyl’s tour.

liftM :: Monad m => (a > b) > m a > m b
. This should be familiar; of course, it is justfmap
. The fact that we have bothfmap
andliftM
is an unfortunate consequence of the fact that theMonad
type class does not require aFunctor
instance, even though mathematically speaking, every monad is a functor. However,fmap
andliftM
are essentially interchangeable, since an instance ofMonad
also has to be instance ofFunctor
.

ap :: Monad m => m (a > b) > m a > m b
should also be familiar: it is equivalent to(<*>)
, justifying the claim that theMonad
interface is strictly more powerful thanApplicative
. We can make anyMonad
into an instance ofApplicative
by settingpure = return
and(<*>) = ap
.

sequence :: Monad m => [m a] > m [a]
takes a list of computations and combines them into one computation which collects a list of their results. It is again something of a historical accident thatsequence
has aMonad
constraint, since it can actually be implemented only in terms ofApplicative
. There is an additional generalization ofsequence
to structures other than lists, which will be discussed in the section onTraversable
.

replicateM :: Monad m => Int > m a > m [a]
is simply a combination ofreplicate
andsequence
.

when :: Monad m => Bool > m () > m ()
conditionally executes a computation, evaluating to its second argument if the test isTrue
, and toreturn ()
if the test isFalse
. A collection of other sorts of monadic conditionals can be found in theIfElse
package.

mapM :: Monad m => (a > m b) > [a] > m [b]
maps its first argument over the second, andsequence
s the results. TheforM
function is justmapM
with its arguments reversed; it is calledforM
since it models generalizedfor
loops: the list[a]
provides the loop indices, and the functiona > m b
specifies the “body” of the loop for each index.

(=<<) :: Monad m => (a > m b) > m a > m b
is just(>>=)
with its arguments reversed; sometimes this direction is more convenient since it corresponds more closely to function application.

(>=>) :: Monad m => (a > m b) > (b > m c) > a > m c
is sort of like function composition, but with an extram
on the result type of each function, and the arguments swapped. We’ll have more to say about this operation later. There is also a flipped variant,(<=<)
.
 The
guard
function is for use with instances ofMonadPlus
, which is discussed at the end of theMonoid
section.
Many of these functions also have “underscored” variants, such as sequence_
and mapM_
; these variants throw away the results of the computations passed to them as arguments, using them only for their side effects.
Other monadic functions which are occasionally useful include filterM
, zipWithM
, foldM
, and forever
.
Laws
There are several laws that instances of Monad
should satisfy (see also the Monad laws wiki page). The standard presentation is:
return a >>= k = k a
m >>= return = m
m >>= (\x > k x >>= h) = (m >>= k) >>= h
fmap f xs = xs >>= return . f = liftM f xs
The first and second laws express the fact that return
behaves nicely: if we inject a value a
into a monadic context with return
, and then bind to k
, it is the same as just applying k
to a
in the first place; if we bind a computation m
to return
, nothing changes. The third law essentially says that (>>=)
is associative, sort of. The last law ensures that fmap
and liftM
are the same for types which are instances of both Functor
and Monad
—which, as already noted, should be every instance of Monad
.
∗ I like to pronounce this operator “fish”.
However, the presentation of the above laws, especially the third, is marred by the asymmetry of (>>=)
. It’s hard to look at the laws and see what they’re really saying. I prefer a much more elegant version of the laws, which is formulated in terms of (>=>)
∗. Recall that (>=>)
“composes” two functions of type a > m b
and b > m c
. You can think of something of type a > m b
(roughly) as a function from a
to b
which may also have some sort of effect in the context corresponding to m
. (>=>)
lets us compose these “effectful functions”, and we would like to know what properties (>=>)
has. The monad laws reformulated in terms of (>=>)
are:
return >=> g = g
g >=> return = g
(g >=> h) >=> k = g >=> (h >=> k)
∗ As fans of category theory will note, these laws say precisely that functions of type a > m b
are the arrows of a category with (>=>)
as composition! Indeed, this is known as the Kleisli category of the monad m
. It will come up again when we discuss Arrow
s.
Ah, much better! The laws simply state that return
is the identity of (>=>)
, and that (>=>)
is associative ∗.
There is also a formulation of the monad laws in terms of fmap
, return
, and join
; for a discussion of this formulation, see the Haskell wikibook page on category theory.
Exercises 


do
notation
Haskell’s special do
notation supports an “imperative style” of programming by providing syntactic sugar for chains of monadic expressions. The genesis of the notation lies in realizing that something like a >>= \x > b >> c >>= \y > d
can be more readably written by putting successive computations on separate lines:
a >>= \x >
b >>
c >>= \y >
d
This emphasizes that the overall computation consists of four computations a
, b
, c
, and d
, and that x
is bound to the result of a
, and y
is bound to the result of c
(b
, c
, and d
are allowed to refer to x
, and d
is allowed to refer to y
as well). From here it is not hard to imagine a nicer notation:
do { x < a
; b
; y < c
; d
}
(The curly braces and semicolons may optionally be omitted; the Haskell parser uses layout to determine where they should be inserted.) This discussion should make clear that do
notation is just syntactic sugar. In fact, do
blocks are recursively translated into monad operations (almost) like this:
do e → e do { e; stmts } → e >> do { stmts } do { v < e; stmts } → e >>= \v > do { stmts } do { let decls; stmts} → let decls in do { stmts }
This is not quite the whole story, since v
might be a pattern instead of a variable. For example, one can write
do (x:xs) < foo
bar x
but what happens if foo
produces an empty list? Well, remember that ugly fail
function in the Monad
type class declaration? That’s what happens. See section 3.14 of the Haskell Report for the full details. See also the discussion of MonadPlus
and MonadZero
in the section on other monoidal classes.
A final note on intuition: do
notation plays very strongly to the “computational context” point of view rather than the “container” point of view, since the binding notation x < m
is suggestive of “extracting” a single x
from m
and doing something with it. But m
may represent some sort of a container, such as a list or a tree; the meaning of x < m
is entirely dependent on the implementation of (>>=)
. For example, if m
is a list, x < m
actually means that x
will take on each value from the list in turn.
Further reading
Philip Wadler was the first to propose using monads to structure functional programs. His paper is still a readable introduction to the subject.
∗ All About Monads, Monads as containers, Understanding monads, The Monadic Way, You Could Have Invented Monads! (And Maybe You Already Have.), there’s a monster in my Haskell!, Understanding Monads. For real., Monads in 15 minutes: Backtracking and Maybe, Monads as computation, Practical Monads
There are, of course, numerous monad tutorials of varying quality ∗.
A few of the best include Cale Gibbard’s Monads as containers and Monads as computation; Jeff Newbern’s All About Monads, a comprehensive guide with lots of examples; and Dan Piponi’s You Could Have Invented Monads!, which features great exercises. If you just want to know how to use IO
, you could consult the Introduction to IO. Even this is just a sampling; the monad tutorials timeline is a more complete list. (All these monad tutorials have prompted parodies like think of a monad ... as well as other kinds of backlash like Monads! (and Why Monad Tutorials Are All Awful) or Abstraction, intuition, and the “monad tutorial fallacy”.)
Other good monad references which are not necessarily tutorials include HenkJan van Tuyl’s tour of the functions in Control.Monad
, Dan Piponi’s field guide, Tim Newsham’s What’s a Monad?, and Chris Smith's excellent article Why Do Monads Matter?. There are also many blog posts which have been written on various aspects of monads; a collection of links can be found under Blog articles/Monads.
For help constructing monads from scratch, and for obtaining a "deep embedding" of monad operations suitable for use in, say, compiling a domainspecific language, see Apfelmus's operational package.
One of the quirks of the Monad
class and the Haskell type system is that it is not possible to straightforwardly declare Monad
instances for types which require a class constraint on their data, even if they are monads from a mathematical point of view. For example, Data.Set
requires an Ord
constraint on its data, so it cannot be easily made an instance of Monad
. A solution to this problem was first described by Eric Kidd, and later made into a library named rmonad by Ganesh Sittampalam and Peter Gavin.
There are many good reasons for eschewing do
notation; some have gone so far as to consider it harmful.
Monads can be generalized in various ways; for an exposition of one possibility, see Robert Atkey’s paper on parameterized monads, or Dan Piponi’s Beyond Monads.
For the categorically inclined, monads can be viewed as monoids (From Monoids to Monads) and also as closure operators Triples and Closure. Derek Elkins’ article in issue 13 of the Monad.Reader contains an exposition of the categorytheoretic underpinnings of some of the standard Monad
instances, such as State
and Cont
. Jonathan Hill and Keith Clarke have an early paper explaining the connection between monads as they arise in category theory and as used in functional programming. There is also a web page by Oleg Kiselyov explaining the history of the IO monad.
Links to many more research papers related to monads can be found under Research papers/Monads and arrows.
Monad transformers
One would often like to be able to combine two monads into one: for example, to have stateful, nondeterministic computations (State
+ []
), or computations which may fail and can consult a readonly environment (Maybe
+ Reader
), and so on. Unfortunately, monads do not compose as nicely as applicative functors (yet another reason to use Applicative
if you don’t need the full power that Monad
provides), but some monads can be combined in certain ways.
Standard monad transformers
The transformers library provides a number of standard monad transformers. Each monad transformer adds a particular capability/feature/effect to any existing monad.

IdentityT
is the identity transformer, which maps a monad to (something isomorphic to) itself. This may seem useless at first glance, but it is useful for the same reason that theid
function is useful  it can be passed as an argument to things which are parameterized over an arbitrary monad transformer, when you do not actually want any extra capabilities. 
StateT
adds a readwrite state. 
ReaderT
adds a readonly environment. 
WriterT
adds a writeonly log. 
RWST
conveniently combinesReaderT
,WriterT
, andStateT
into one. 
MaybeT
adds the possibility of failure. 
ErrorT
adds the possibility of failure with an arbitrary type to represent errors. 
ListT
adds nondeterminism (however, see the discussion ofListT
below). 
ContT
adds continuation handling.
For example, StateT s Maybe
is an instance of Monad
; computations of type StateT s Maybe a
may fail, and have access to a mutable state of type s
. Monad transformers can be multiply stacked. One thing to keep in mind while using monad transformers is that the order of composition matters. For example, when a StateT s Maybe a
computation fails, the state ceases being updated (indeed, it simply disappears); on the other hand, the state of a MaybeT (State s) a
computation may continue to be modified even after the computation has "failed". This may seem backwards, but it is correct. Monad transformers build composite monads “inside out”; MaybeT (State s) a
is isomorphic to s > (Maybe a, s)
. (Lambdabot has an indispensable @unmtl
command which you can use to “unpack” a monad transformer stack in this way.)
Intuitively, the monads become "more fundamental" the further inside the stack you get, and the effects of inner monads "have precedence" over the effects of outer ones. Of course, this is just handwaving, and if you are unsure of the proper order for some monads you wish to combine, there is no substitute for using @unmtl
or simply trying out the various options.
Definition and laws
All monad transformers should implement the MonadTrans
type class, defined in Control.Monad.Trans.Class
:
class MonadTrans t where
lift :: Monad m => m a > t m a
It allows arbitrary computations in the base monad m
to be “lifted” into computations in the transformed monad t m
. (Note that type application associates to the left, just like function application, so t m a = (t m) a
.)
lift
must satisfy the laws
lift . return = return
lift (m >>= f) = lift m >>= (lift . f)
which intuitively state that lift
transforms m a
computations into t m a
computations in a "sensible" way, which sends the return
and (>>=)
of m
to the return
and (>>=)
of t m
.
Exercises 


Transformer type classes and "capability" style
∗ The only problem with this scheme is the quadratic number of instances required as the number of standard monad transformers grows—but as the current set of standard monad transformers seems adequate for most common use cases, this may not be that big of a deal.
There are also type classes (provided by the mtl
package) for the operations of each transformer. For example, the MonadState
type class provides the statespecific methods get
and put
, allowing you to conveniently use these methods not only with State
, but with any monad which is an instance of MonadState
—including MaybeT (State s)
, StateT s (ReaderT r IO)
, and so on. Similar type classes exist for Reader
, Writer
, Cont
, IO
, and others ∗.
These type classes serve two purposes. First, they get rid of (most of) the need for explicitly using lift
, giving a typedirected way to automatically determine the right number of calls to lift
. Simply writing put
will be automatically translated into lift . put
, lift . lift . put
, or something similar depending on what concrete monad stack you are using.
Second, they give you more flexibility to switch between different concrete monad stacks. For example, if you are writing a statebased algorithm, don't write
foo :: State Int Char
foo = modify (*2) >> return 'x'
but rather
foo :: MonadState Int m => m Char
foo = modify (*2) >> return 'x'
Now, if somewhere down the line you realize you need to introduce the possibility of failure, you might switch from State Int
to MaybeT (State Int)
. The type of the first version of foo
would need to be modified to reflect this change, but the second version of foo
can still be used asis.
However, this sort of "capabilitybased" style (e.g. specifying that foo
works for any monad with the "state capability") quickly runs into problems when you try to naively scale it up: for example, what if you need to maintain two independent states? A framework for solving this and related problems is described by Schrijvers and Olivera (Monads, zippers and views: virtualizing the monad stack, ICFP 2011) and is implemented in the Monatron
package.
Composing monads
Is the composition of two monads always a monad? As hinted previously, the answer is no.
Since Applicative
functors are closed under composition, the problem must lie with join
. Indeed, suppose m
and n
are arbitrary monads; to make a monad out of their composition we would need to be able to implement
join :: m (n (m (n a))) > m (n a)
but it is not clear how this could be done in general. The join
method for m
is no help, because the two occurrences of m
are not next to each other (and likewise for n
).
However, one situation in which it can be done is if n
distributes over m
, that is, if there is a function
distrib :: n (m a) > m (n a)
satisfying certain laws. See Jones and Duponcheel (Composing Monads); see also the section on Traversable.
For a much more indepth discussion and analysis of the failure of monads to be closed under composition, see this question on StackOverflow.
Exercises 


Further reading
Much of the monad transformer library (originally mtl
, now split between mtl
and transformers
), including the Reader
, Writer
, State
, and other monads, as well as the monad transformer framework itself, was inspired by Mark Jones’ classic paper Functional Programming with Overloading and HigherOrder Polymorphism. It’s still very much worth a read—and highly readable—after almost fifteen years.
See Edward Kmett's mailing list message for a description of the history and relationships among monad transformer packages (mtl
, transformers
, monadsfd
, monadstf
).
There are two excellent references on monad transformers. Martin Grabmüller’s Monad Transformers Step by Step is a thorough description, with running examples, of how to use monad transformers to elegantly build up computations with various effects. Cale Gibbard’s article on how to use monad transformers is more practical, describing how to structure code using monad transformers to make writing it as painless as possible. Another good starting place for learning about monad transformers is a blog post by Dan Piponi.
The ListT
transformer from the transformers
package comes with the caveat that ListT m
is only a monad when m
is commutative, that is, when ma >>= \a > mb >>= \b > foo
is equivalent to mb >>= \b > ma >>= \a > foo
(i.e. the order of m
's effects does not matter). For one explanation why, see Dan Piponi's blog post "Why isn't ListT []
a monad". For more examples, as well as a design for a version of ListT
which does not have this problem, see ListT
done right.
There is an alternative way to compose monads, using coproducts, as described by Lüth and Ghani. This method is interesting but has not (yet?) seen widespread use. For a more recent alternative, see Kiselyov et al's Extensible Effects: An Alternative to Monad Transformers.
MonadFix
Note: MonadFix
is included here for completeness (and because it is interesting) but seems not to be used much. Skipping this section on a first readthrough is perfectly OK (and perhaps even recommended).
mdo
/do rec
notation
∗ In GHC 7.6, the flag has been changed to XRecursiveDo
.
The MonadFix
class describes monads which support the special fixpoint operation mfix :: (a > m a) > m a
, which allows the output of monadic computations to be defined via (effectful) recursion. This is supported in GHC by a special “recursive do” notation, enabled by the XDoRec
flag∗. Within a do
block, one may have a nested rec
block, like so:
do { x < foo
; rec { y < baz
; z < bar
; bob
}
; w < frob
}
Normally (if we had do
in place of rec
in the above example), y
would be in scope in bar
and bob
but not in baz
, and z
would be in scope only in bob
. With the rec
, however, y
and z
are both in scope in all three of baz
, bar
, and bob
. A rec
block is analogous to a let
block such as
let { y = baz
; z = bar
}
in bob
because, in Haskell, every variable bound in a let
block is in scope throughout the entire block. (From this point of view, Haskell's normal do
blocks are analogous to Scheme's let*
construct.)
What could such a feature be used for? One of the motivating examples given in the original paper describing MonadFix
(see below) is encoding circuit descriptions. A line in a do
block such as
x < gate y z
describes a gate whose input wires are labeled y
and z
and whose output wire is labeled x
. Many (most?) useful circuits, however, involve some sort of feedback loop, making them impossible to write in a normal do
block (since some wire would have to be mentioned as an input before being listed as an output). Using a rec
block solves this problem.
Examples and intuition
Of course, not every monad supports such recursive binding. However, as mentioned above, it suffices to have an implementation of mfix :: (a > m a) > m a
, satisfying a few laws. Let's try implementing mfix
for the Maybe
monad. That is, we want to implement a function
maybeFix :: (a > Maybe a) > Maybe a
∗ Actually, fix
is implemented slightly differently for efficiency reasons; but the given definition is equivalent and simpler for the present purpose.
Let's think for a moment about the implementation ∗ of the nonmonadic fix :: (a > a) > a
:
fix f = f (fix f)
Inspired by fix
, our first attempt at implementing maybeFix
might be something like
maybeFix :: (a > Maybe a) > Maybe a
maybeFix f = maybeFix f >>= f
This has the right type. However, something seems wrong: there is nothing in particular here about Maybe
; maybeFix
actually has the more general type Monad m => (a > m a) > m a
. But didn't we just say that not all monads support mfix
?
The answer is that although this implementation of maybeFix
has the right type, it does not have the intended semantics. If we think about how (>>=)
works for the Maybe
monad (by patternmatching on its first argument to see whether it is Nothing
or Just
) we can see that this definition of maybeFix
is completely useless: it will just recurse infinitely, trying to decide whether it is going to return Nothing
or Just
, without ever even so much as a glance in the direction of f
.
The trick is to simply assume that maybeFix
will return Just
, and get on with life!
maybeFix :: (a > Maybe a) > Maybe a
maybeFix f = ma
where ma = f (fromJust ma)
This says that the result of maybeFix
is ma
, and assuming that ma = Just x
, it is defined (recursively) to be equal to f x
.
Why is this OK? Isn't fromJust
almost as bad as unsafePerformIO
? Well, usually, yes. This is just about the only situation in which it is justified! The interesting thing to note is that maybeFix
will never crash  although it may, of course, fail to terminate. The only way we could get a crash is if we try to evaluate fromJust ma
when we know that ma = Nothing
. But how could we know ma = Nothing
? Since ma
is defined as f (fromJust ma)
, it must be that this expression has already been evaluated to Nothing
 in which case there is no reason for us to be evaluating fromJust ma
in the first place!
To see this from another point of view, we can consider three possibilities. First, if f
outputs Nothing
without looking at its argument, then maybeFix f
clearly returns Nothing
. Second, if f
always outputs Just x
, where x
depends on its argument, then the recursion can proceed usefully: fromJust ma
will be able to evaluate to x
, thus feeding f
's output back to it as input. Third, if f
tries to use its argument to decide whether to output Just
or Nothing
, then maybeFix f
will not terminate: evaluating f
's argument requires evaluating ma
to see whether it is Just
, which requires evaluating f (fromJust ma)
, which requires evaluating ma
, ... and so on.
There are also instances of MonadFix
for lists (which works analogously to the instance for Maybe
), for ST
, and for IO
. The instance for IO
is particularly amusing: it creates a new (empty) MVar
, immediately reads its contents using unsafeInterleaveIO
(which delays the actual reading lazily until the value is needed), uses the contents of the MVar
to compute a new value, which it then writes back into the MVar
. It almost seems, spookily, that mfix
is sending a value back in time to itself through the MVar
 though of course what is really going on is that the reading is delayed just long enough (via unsafeInterleaveIO
) to get the process bootstrapped.
Exercises 


GHC 7.6 changes
GHC 7.6 reinstated the old mdo
syntax, so the example at the start of this section can be written
mdo { x < foo
; y < baz
; z < bar
; bob
; w < frob
}
which will be translated into the original example (assuming that, say, bar
and bob
refer to y
. The difference is that mdo
will analyze the code in order to find minimal recursive blocks, which will be placed in rec
blocks, whereas rec
blocks desugar directly into calls to mfix
without any further analysis.
Further reading
For more information (such as the precise desugaring rules for rec
blocks), see Levent Erkök and John Launchbury's 2002 Haskell workshop paper, A Recursive do for Haskell, or for full details, Levent Erkök’s thesis, Value Recursion in Monadic Computations. (Note, while reading, that MonadFix
used to be called MonadRec
.) You can also read the GHC user manual section on recursive donotation.
Semigroup
A semigroup is a set together with a binary operation which combines elements from . The operator is required to be associative (that is, , for any which are elements of ).
For example, the natural numbers under addition form a semigroup: the sum of any two natural numbers is a natural number, and for any natural numbers , , and . The integers under multiplication also form a semigroup, as do the integers (or rationals, or reals) under or , Boolean values under conjunction and disjunction, lists under concatenation, functions from a set to itself under composition ... Semigroups show up all over the place, once you know to look for them.
Definition
Semigroups are not (yet?) defined in the base package, but the semigroups package provides a standard definition.
The definition of the Semigroup
type class (haddock) is as follows:
class Semigroup a where
(<>) :: a > a > a
sconcat :: NonEmpty a > a
sconcat = sconcat (a : as) = go a as where
go b (c:cs) = b <> go c cs
go b [] = b
times1p :: Whole n => n > a > a
times1p = ...
The really important method is (<>)
, representing the associative binary operation. The other two methods have default implementations in terms of (<>)
, and are included in the type class in case some instances can give more efficient implementations than the default. sconcat
reduces a nonempty list using (<>)
; times1p n
is equivalent to (but more efficient than) sconcat . replicate n
. See the haddock documentation for more information on sconcat
and times1p
.
Laws
The only law is that (<>)
must be associative:
(x <> y) <> z = x <> (y <> z)
Monoid
Many semigroups have a special element for which the binary operation is the identity, that is, for every element . Such a semigroupwithidentityelement is called a monoid.
Definition
The definition of the Monoid
type class (defined in
Data.Monoid
; haddock) is:
class Monoid a where
mempty :: a
mappend :: a > a > a
mconcat :: [a] > a
mconcat = foldr mappend mempty
The mempty
value specifies the identity element of the monoid, and mappend
is the binary operation. The default definition for mconcat
“reduces” a list of elements by combining them all with mappend
,
using a right fold. It is only in the Monoid
class so that specific
instances have the option of providing an alternative, more efficient
implementation; usually, you can safely ignore mconcat
when creating
a Monoid
instance, since its default definition will work just fine.
The Monoid
methods are rather unfortunately named; they are inspired
by the list instance of Monoid
, where indeed mempty = []
and mappend = (++)
, but this is misleading since many
monoids have little to do with appending (see these Comments from OCaml Hacker Brian Hurt on the Haskellcafe mailing list). This was improved in GHC 7.4, where (<>)
was added as an alias to mappend
.
Laws
Of course, every Monoid
instance should actually be a monoid in the
mathematical sense, which implies these laws:
mempty `mappend` x = x
x `mappend` mempty = x
(x `mappend` y) `mappend` z = x `mappend` (y `mappend` z)
Instances
There are quite a few interesting Monoid
instances defined in Data.Monoid
.
[a]
is aMonoid
, withmempty = []
andmappend = (++)
. It is not hard to check that(x ++ y) ++ z = x ++ (y ++ z)
for any listsx
,y
, andz
, and that the empty list is the identity:[] ++ x = x ++ [] = x
. As noted previously, we can make a monoid out of any numeric type under either addition or multiplication. However, since we can’t have two instances for the same type,
Data.Monoid
provides twonewtype
wrappers,Sum
andProduct
, with appropriateMonoid
instances.> getSum (mconcat . map Sum $ [1..5]) 15 > getProduct (mconcat . map Product $ [1..5]) 120
This example code is silly, of course; we could just write
sum [1..5]
andproduct [1..5]
. Nevertheless, these instances are useful in more generalized settings, as we will see in the section onFoldable
. Any
andAll
arenewtype
wrappers providingMonoid
instances forBool
(under disjunction and conjunction, respectively). There are three instances for
Maybe
: a basic instance which lifts aMonoid
instance fora
to an instance forMaybe a
, and twonewtype
wrappersFirst
andLast
for whichmappend
selects the first (respectively last) nonNothing
item. Endo a
is a newtype wrapper for functionsa > a
, which form a monoid under composition. There are several ways to “lift”
Monoid
instances to instances with additional structure. We have already seen that an instance fora
can be lifted to an instance forMaybe a
. There are also tuple instances: ifa
andb
are instances ofMonoid
, then so is(a,b)
, using the monoid operations fora
andb
in the obvious pairwise manner. Finally, ifa
is aMonoid
, then so is the function typee > a
for anye
; in particular,g `mappend` h
is the function which applies bothg
andh
to its argument and then combines the results using the underlyingMonoid
instance fora
. This can be quite useful and elegant (see example).  The type
Ordering = LT  EQ  GT
is aMonoid
, defined in such a way thatmconcat (zipWith compare xs ys)
computes the lexicographic ordering ofxs
andys
(ifxs
andys
have the same length). In particular,mempty = EQ
, andmappend
evaluates to its leftmost nonEQ
argument (orEQ
if both arguments areEQ
). This can be used together with the function instance ofMonoid
to do some clever things (example).  There are also
Monoid
instances for several standard data structures in the containers library (haddock), includingMap
,Set
, andSequence
.
Monoid
is also used to enable several other type class instances.
As noted previously, we can use Monoid
to make ((,) e)
an instance of Applicative
:
instance Monoid e => Applicative ((,) e) where
pure x = (mempty, x)
(u, f) <*> (v, x) = (u `mappend` v, f x)
Monoid
can be similarly used to make ((,) e)
an instance of Monad
as well; this is known as the writer monad. As we’ve already seen, Writer
and WriterT
are a newtype wrapper and transformer for this monad, respectively.
Monoid
also plays a key role in the Foldable
type class (see section Foldable).
Other monoidal classes: Alternative, MonadPlus, ArrowPlus
The Alternative
type class (haddock)
is for Applicative
functors which also have
a monoid structure:
class Applicative f => Alternative f where
empty :: f a
(<>) :: f a > f a > f a
Of course, instances of Alternative
should satisfy the monoid laws
empty <> x = x
x <> empty = x
(x <> y) <> z = x <> (y <> z)
Likewise, MonadPlus
(haddock)
is for Monad
s with a monoid structure:
class Monad m => MonadPlus m where
mzero :: m a
mplus :: m a > m a > m a
The MonadPlus
documentation states that it is intended to model
monads which also support “choice and failure”; in addition to the
monoid laws, instances of MonadPlus
are expected to satisfy
mzero >>= f = mzero
v >> mzero = mzero
which explains the sense in which mzero
denotes failure. Since
mzero
should be the identity for mplus
, the computation m1 `mplus` m2
succeeds (evaluates to something other than mzero
) if
either m1
or m2
does; so mplus
represents choice. The guard
function can also be used with instances of MonadPlus
; it requires a
condition to be satisfied and fails (using mzero
) if it is not. A
simple example of a MonadPlus
instance is []
, which is exactly the
same as the Monoid
instance for []
: the empty list represents
failure, and list concatenation represents choice. In general,
however, a MonadPlus
instance for a type need not be the same as its
Monoid
instance; Maybe
is an example of such a type. A great
introduction to the MonadPlus
type class, with interesting examples
of its use, is Doug Auclair’s MonadPlus: What a Super Monad! in the Monad.Reader issue 11.
There used to be a type class called MonadZero
containing only
mzero
, representing monads with failure. The do
notation requires
some notion of failure to deal with failing pattern matches.
Unfortunately, MonadZero
was scrapped in favor of adding the fail
method to the Monad
class. If we are lucky, someday MonadZero
will
be restored, and fail
will be banished to the bit bucket where it
belongs (see MonadPlus reform proposal). The idea is that any
do
block which uses pattern matching (and hence may fail) would require
a MonadZero
constraint; otherwise, only a Monad
constraint would be
required.
Finally, ArrowZero
and ArrowPlus
(haddock)
represent Arrow
s (see below) with a
monoid structure:
class Arrow arr => ArrowZero arr where
zeroArrow :: b `arr` c
class ArrowZero arr => ArrowPlus arr where
(<+>) :: (b `arr` c) > (b `arr` c) > (b `arr` c)
Further reading
Monoids have gotten a fair bit of attention recently, ultimately due
to
a blog post by Brian Hurt, in which he
complained about the fact that the names of many Haskell type classes
(Monoid
in particular) are taken from abstract mathematics. This
resulted in a long Haskellcafe thread
arguing the point and discussing monoids in general.
∗ May its name live forever.
However, this was quickly followed by several blog posts about
Monoid
∗. First, Dan Piponi
wrote a great introductory post, Haskell Monoids and their Uses. This was quickly followed by
Heinrich Apfelmus’ Monoids and Finger Trees, an accessible exposition of
Hinze and Paterson’s classic paper on 23 finger trees, which makes very clever
use of Monoid
to implement an elegant and generic data structure.
Dan Piponi then wrote two fascinating articles about using Monoids
(and finger trees): Fast Incremental Regular Expressions and Beyond Regular Expressions
In a similar vein, David Place’s article on improving Data.Map
in
order to compute incremental folds (see the Monad Reader issue 11)
is also a
good example of using Monoid
to generalize a data structure.
Some other interesting examples of Monoid
use include building elegant list sorting combinators, collecting unstructured information, combining probability distributions, and a brilliant series of posts by ChungChieh Shan and Dylan Thurston using Monoid
s to elegantly solve a difficult combinatorial puzzle (followed by part 2, part 3, part 4).
As unlikely as it sounds, monads can actually be viewed as a sort of
monoid, with join
playing the role of the binary operation and
return
the role of the identity; see Dan Piponi’s blog post.
Foldable
The Foldable
class, defined in the Data.Foldable
module (haddock), abstracts over containers which can be
“folded” into a summary value. This allows such folding operations
to be written in a containeragnostic way.
Definition
The definition of the Foldable
type class is:
class Foldable t where
fold :: Monoid m => t m > m
foldMap :: Monoid m => (a > m) > t a > m
foldr :: (a > b > b) > b > t a > b
foldl :: (a > b > a) > a > t b > a
foldr1 :: (a > a > a) > t a > a
foldl1 :: (a > a > a) > t a > a
This may look complicated, but in fact, to make a Foldable
instance
you only need to implement one method: your choice of foldMap
or
foldr
. All the other methods have default implementations in terms
of these, and are presumably included in the class in case more
efficient implementations can be provided.
Instances and examples
The type of foldMap
should make it clear what it is supposed to do:
given a way to convert the data in a container into a Monoid
(a
function a > m
) and a container of a
’s (t a
), foldMap
provides a way to iterate over the entire contents of the container,
converting all the a
’s to m
’s and combining all the m
’s with
mappend
. The following code shows two examples: a simple
implementation of foldMap
for lists, and a binary tree example
provided by the Foldable
documentation.
instance Foldable [] where
foldMap g = mconcat . map g
data Tree a = Empty  Leaf a  Node (Tree a) a (Tree a)
instance Foldable Tree where
foldMap f Empty = mempty
foldMap f (Leaf x) = f x
foldMap f (Node l k r) = foldMap f l `mappend` f k `mappend` foldMap f r
The foldr
function has a type similar to the foldr
found in the Prelude
, but
more general, since the foldr
in the Prelude
works only on lists.
The Foldable
module also provides instances for Maybe
and Array
;
additionally, many of the data structures found in the standard containers library (for example, Map
, Set
, Tree
,
and Sequence
) provide their own Foldable
instances.
Exercises 


Derived folds
Given an instance of Foldable
, we can write generic,
containeragnostic functions such as:
 Compute the size of any container.
containerSize :: Foldable f => f a > Int
containerSize = getSum . foldMap (const (Sum 1))
 Compute a list of elements of a container satisfying a predicate.
filterF :: Foldable f => (a > Bool) > f a > [a]
filterF p = foldMap (\a > if p a then [a] else [])
 Get a list of all the Strings in a container which include the
 letter a.
aStrings :: Foldable f => f String > [String]
aStrings = filterF (elem 'a')
The Foldable
module also provides a large number of predefined
folds, many of which are generalized versions of Prelude
functions of the
same name that only work on lists: concat
, concatMap
, and
,
or
, any
, all
, sum
, product
, maximum
(By
),
minimum
(By
), elem
, notElem
, and find
.
The important function toList
is also provided, which turns any Foldable
structure into a list of its elements in leftright order; it works by folding with the list monoid.
There are also generic functions that work with Applicative
or
Monad
instances to generate some sort of computation from each
element in a container, and then perform all the side effects from
those computations, discarding the results: traverse_
, sequenceA_
,
and others. The results must be discarded because the Foldable
class is too weak to specify what to do with them: we cannot, in
general, make an arbitrary Applicative
or Monad
instance into a Monoid
, but we can make m ()
into a Monoid
for any such m
. If we do have an Applicative
or Monad
with a monoid
structure—that is, an Alternative
or a MonadPlus
—then we can
use the asum
or msum
functions, which can combine the results as
well. Consult the Foldable
documentation for
more details on any of these functions.
Note that the Foldable
operations always forget the structure of
the container being folded. If we start with a container of type t a
for some Foldable t
, then t
will never appear in the output
type of any operations defined in the Foldable
module. Many times
this is exactly what we want, but sometimes we would like to be able
to generically traverse a container while preserving its
structure—and this is exactly what the Traversable
class provides,
which will be discussed in the next section.
Exercises 


Foldable actually isn't
The generic term "fold" is often used to refer to the more technical concept of catamorphism. Intuitively, given a way to summarize "one level of structure" (where recursive subterms have already been replaced with their summaries), a catamorphism can summarize an entire recursive structure. It is important to realize that Foldable
does not correspond to catamorphisms, but to something weaker. In particular, Foldable
allows observing only the leftright order of elements within a structure, not the actual structure itself. Put another way, every use of Foldable
can be expressed in terms of toList
. For example, fold
itself is equivalent to mconcat . toList
.
This is sufficient for many tasks, but not all. For example, consider trying to compute the depth of a Tree
: try as we might, there is no way to implement it using Foldable
. However, it can be implemented as a catamorphism.
Further reading
The Foldable
class had its genesis in McBride and Paterson’s paper
introducing Applicative
, although it has
been fleshed out quite a bit from the form in the paper.
An interesting use of Foldable
(as well as Traversable
) can be
found in Janis Voigtländer’s paper Bidirectionalization for free!.
Traversable
Definition
The Traversable
type class, defined in the Data.Traversable
module (haddock), is:
class (Functor t, Foldable t) => Traversable t where
traverse :: Applicative f => (a > f b) > t a > f (t b)
sequenceA :: Applicative f => t (f a) > f (t a)
mapM :: Monad m => (a > m b) > t a > m (t b)
sequence :: Monad m => t (m a) > m (t a)
As you can see, every Traversable
is also a foldable functor. Like
Foldable
, there is a lot in this type class, but making instances is
actually rather easy: one need only implement traverse
or
sequenceA
; the other methods all have default implementations in
terms of these functions. A good exercise is to figure out what the default
implementations should be: given either traverse
or sequenceA
, how
would you define the other three methods? (Hint for mapM
:
Control.Applicative
exports the WrapMonad
newtype, which makes any
Monad
into an Applicative
. The sequence
function can be implemented in terms
of mapM
.)
Intuition
The key method of the Traversable
class, and the source of its
unique power, is sequenceA
. Consider its type:
sequenceA :: Applicative f => t (f a) > f (t a)
This answers the fundamental question: when can we commute two functors? For example, can we turn a tree of lists into a list of trees?
The ability to compose two monads depends crucially on this ability to
commute functors. Intuitively, if we want to build a composed monad
M a = m (n a)
out of monads m
and n
, then to be able to
implement join :: M (M a) > M a
, that is,
join :: m (n (m (n a))) > m (n a)
, we have to be able to commute
the n
past the m
to get m (m (n (n a)))
, and then we can use the
join
s for m
and n
to produce something of type m (n a)
. See
Mark Jones’ paper for more details.
Alternatively, looking at the type of traverse
,
traverse :: Applicative f => (a > f b) > t a > f (t b)
leads us to view Traversable
as a generalization of Functor
. traverse
is an "effectful fmap
": it allows us to map over a structure of type t a
, applying a function to every element of type a
and in order to produce a new structure of type t b
; but along the way the function may have some effects (captured by the applicative functor f
).
Exercises 


Instances and examples
What’s an example of a Traversable
instance?
The following code shows an example instance for the same
Tree
type used as an example in the previous Foldable
section. It
is instructive to compare this instance with a Functor
instance for
Tree
, which is also shown.
data Tree a = Empty  Leaf a  Node (Tree a) a (Tree a)
instance Traversable Tree where
traverse g Empty = pure Empty
traverse g (Leaf x) = Leaf <$> g x
traverse g (Node l x r) = Node <$> traverse g l
<*> g x
<*> traverse g r
instance Functor Tree where
fmap g Empty = Empty
fmap g (Leaf x) = Leaf $ g x
fmap g (Node l x r) = Node (fmap g l)
(g x)
(fmap g r)
It should be clear that the Traversable
and Functor
instances for
Tree
are almost identical; the only difference is that the Functor
instance involves normal function application, whereas the
applications in the Traversable
instance take place within an
Applicative
context, using (<$>)
and (<*>)
. In fact, this will
be
true for any type.
Any Traversable
functor is also Foldable
, and a Functor
. We can see
this not only from the class declaration, but by the fact that we can
implement the methods of both classes given only the Traversable
methods.
The standard libraries provide a number of Traversable
instances,
including instances for []
, Maybe
, Map
, Tree
, and Sequence
.
Notably, Set
is not Traversable
, although it is Foldable
.
Exercises 


Laws
Any instance of Traversable
must satisfy the following two laws, where Identity
is the identity functor (as defined in the Data.Functor.Identity
module from the transformers
package), and Compose
wraps the composition of two functors (as defined in Data.Functor.Compose
):

traverse Identity = Identity

traverse (Compose . fmap g . f) = Compose . fmap (traverse g) . traverse f
The first law essentially says that traversals cannot make up arbitrary effects. The second law explains how doing two traversals in sequence can be collapsed to a single traversal.
Additionally, suppose eta
is an "Applicative
morphism", that is,
eta :: forall a f g. (Applicative f, Applicative g) => f a > g a
and eta
preserves the Applicative
operations: eta (pure x) = pure x
and eta (x <*> y) = eta x <*> eta y
. Then, by parametricity, any instance of Traversable
satisfying the above two laws will also satisfy eta . traverse f = traverse (eta . f)
.
Further reading
The Traversable
class also had its genesis in McBride and Paterson’s Applicative
paper,
and is described in more detail in Gibbons and Oliveira, The Essence of the Iterator Pattern,
which also contains a wealth of references to related work.
Traversable
forms a core component of Edward Kmett's lens library. Watching Edward's talk on the subject is a highly recommended way to gain better insight into Traversable
, Foldable
, Applicative
, and many other things besides.
For references on the Traversable
laws, see Russell O'Connor's mailing list post (and subsequent thread).
Category
Category
is a relatively recent addition to the Haskell standard libraries. It generalizes the notion of function composition to general “morphisms”.
∗ GHC 7.6.1 changed its rules regarding types and type variables. Now, any operator at the type level is treated as a type constructor rather than a type variable; prior to GHC 7.6.1 it was possible to use (~>)
instead of `arr`
. For more information, see the discussion on the GHCusers mailing list. For a new approach to nice arrow notation that works with GHC 7.6.1, see this message and also this message from Edward Kmett, though for simplicity I haven't adopted it here.
The definition of the Category
type class (from
Control.Category
; haddock) is shown below. For ease of reading, note that I have used an infix type variable `arr`
, in parallel with the infix function type constructor (>)
. ∗ This syntax is not part of Haskell 2010. The second definition shown is the one used in the standard libraries. For the remainder of this document, I will use the infix type constructor `arr`
for Category
as well as Arrow
.
class Category arr where
id :: a `arr` a
(.) :: (b `arr` c) > (a `arr` b) > (a `arr` c)
 The same thing, with a normal (prefix) type constructor
class Category cat where
id :: cat a a
(.) :: cat b c > cat a b > cat a c
Note that an instance of Category
should be a type constructor which takes two type arguments, that is, something of kind * > * > *
. It is instructive to imagine the type constructor variable cat
replaced by the function constructor (>)
: indeed, in this case we recover precisely the familiar identity function id
and function composition operator (.)
defined in the standard Prelude
.
Of course, the Category
module provides exactly such an instance of
Category
for (>)
. But it also provides one other instance, shown below, which should be familiar from the previous discussion of the Monad
laws. Kleisli m a b
, as defined in the Control.Arrow
module, is just a newtype
wrapper around a > m b
.
newtype Kleisli m a b = Kleisli { runKleisli :: a > m b }
instance Monad m => Category (Kleisli m) where
id = Kleisli return
Kleisli g . Kleisli h = Kleisli (h >=> g)
The only law that Category
instances should satisfy is that id
and (.)
should form a monoid—that is, id
should be the identity of (.)
, and (.)
should be associative.
Finally, the Category
module exports two additional operators:
(<<<)
, which is just a synonym for (.)
, and (>>>)
, which is (.)
with its arguments reversed. (In previous versions of the libraries, these operators were defined as part of the Arrow
class.)
Further reading
The name Category
is a bit misleading, since the Category
class cannot represent arbitrary categories, but only categories whose objects are objects of Hask
, the category of Haskell types. For a more general treatment of categories within Haskell, see the categoryextras package. For more about category theory in general, see the excellent Haskell wikibook page,
Steve Awodey’s new book, Benjamin Pierce’s Basic category theory for computer scientists, or Barr and Wells category theory lecture notes. Benjamin Russell’s blog post
is another good source of motivation and category theory links. You certainly don’t need to know any category theory to be a successful and productive Haskell programmer, but it does lend itself to much deeper appreciation of Haskell’s underlying theory.
Arrow
The Arrow
class represents another abstraction of computation, in a
similar vein to Monad
and Applicative
. However, unlike Monad
and Applicative
, whose types only reflect their output, the type of
an Arrow
computation reflects both its input and output. Arrows
generalize functions: if arr
is an instance of Arrow
, a value of
type b `arr` c
can be thought of as a computation which takes values of
type b
as input, and produces values of type c
as output. In the
(>)
instance of Arrow
this is just a pure function; in general, however,
an arrow may represent some sort of “effectful” computation.
Definition
The definition of the Arrow
type class, from
Control.Arrow
(haddock), is:
class Category arr => Arrow arr where
arr :: (b > c) > (b `arr` c)
first :: (b `arr` c) > ((b, d) `arr` (c, d))
second :: (b `arr` c) > ((d, b) `arr` (d, c))
(***) :: (b `arr` c) > (b' `arr` c') > ((b, b') `arr` (c, c'))
(&&&) :: (b `arr` c) > (b `arr` c') > (b `arr` (c, c'))
∗ In versions of the base
package prior to version 4, there is no Category
class, and the
Arrow
class includes the arrow composition operator (>>>)
. It
also includes pure
as a synonym for arr
, but this was removed
since it conflicts with the pure
from Applicative
.
The first thing to note is the Category
class constraint, which
means that we get identity arrows and arrow composition for free:
given two arrows g :: b `arr` c
and h :: c `arr` d
, we can form their
composition g >>> h :: b `arr` d
∗.
As should be a familiar pattern by now, the only methods which must be
defined when writing a new instance of Arrow
are arr
and first
;
the other methods have default definitions in terms of these, but are
included in the Arrow
class so that they can be overridden with more
efficient implementations if desired.
Intuition
Let’s look at each of the arrow methods in turn. Ross Paterson’s web page on arrows has nice diagrams which can help build intuition.
 The
arr
function takes any functionb > c
and turns it into a generalized arrowb `arr` c
. Thearr
method justifies the claim that arrows generalize functions, since it says that we can treat any function as an arrow. It is intended that the arrowarr g
is “pure” in the sense that it only computesg
and has no “effects” (whatever that might mean for any particular arrow type).
 The
first
method turns any arrow fromb
toc
into an arrow from(b,d)
to(c,d)
. The idea is thatfirst g
usesg
to process the first element of a tuple, and lets the second element pass through unchanged. For the function instance ofArrow
, of course,first g (x,y) = (g x, y)
.
 The
second
function is similar tofirst
, but with the elements of the tuples swapped. Indeed, it can be defined in terms offirst
using an auxiliary functionswap
, defined byswap (x,y) = (y,x)
.
 The
(***)
operator is “parallel composition” of arrows: it takes two arrows and makes them into one arrow on tuples, which has the behavior of the first arrow on the first element of a tuple, and the behavior of the second arrow on the second element. The mnemonic is thatg *** h
is the product (hence*
) ofg
andh
. For the function instance ofArrow
, we define(g *** h) (x,y) = (g x, h y)
. The default implementation of(***)
is in terms offirst
,second
, and sequential arrow composition(>>>)
. The reader may also wish to think about how to implementfirst
andsecond
in terms of(***)
.
 The
(&&&)
operator is “fanout composition” of arrows: it takes two arrowsg
andh
and makes them into a new arrowg &&& h
which supplies its input as the input to bothg
andh
, returning their results as a tuple. The mnemonic is thatg &&& h
performs bothg
andh
(hence&
) on its input. For functions, we define(g &&& h) x = (g x, h x)
.
Instances
The Arrow
library itself only provides two Arrow
instances, both
of which we have already seen: (>)
, the normal function
constructor, and Kleisli m
, which makes functions of
type a > m b
into Arrow
s for any Monad m
. These instances are:
instance Arrow (>) where
arr g = g
first g (x,y) = (g x, y)
newtype Kleisli m a b = Kleisli { runKleisli :: a > m b }
instance Monad m => Arrow (Kleisli m) where
arr f = Kleisli (return . f)
first (Kleisli f) = Kleisli (\ ~(b,d) > do c < f b
return (c,d) )
Laws
∗ See John Hughes: Generalising monads to arrows; Sam Lindley, Philip Wadler, Jeremy Yallop: The arrow calculus; Ross Paterson: Programming with Arrows.
There are quite a few laws that instances of Arrow
should
satisfy ∗:
arr id = id
arr (h . g) = arr g >>> arr h
first (arr g) = arr (g *** id)
first (g >>> h) = first g >>> first h
first g >>> arr (id *** h) = arr (id *** h) >>> first g
first g >>> arr fst = arr fst >>> g
first (first g) >>> arr assoc = arr assoc >>> first g
assoc ((x,y),z) = (x,(y,z))
Note that this version of the laws is slightly different than the laws given in the
first two above references, since several of the laws have now been
subsumed by the Category
laws (in particular, the requirements that
id
is the identity arrow and that (>>>)
is associative). The laws
shown here follow those in Paterson’s Programming with Arrows, which uses the
Category
class.
∗ Unless categorytheoryinduced insomnolence is your cup of tea.
The reader is advised not to lose too much sleep over the Arrow
laws ∗, since it is not essential to understand them in order to
program with arrows. There are also laws that ArrowChoice
,
ArrowApply
, and ArrowLoop
instances should satisfy; the interested
reader should consult Paterson: Programming with Arrows.
ArrowChoice
Computations built using the Arrow
class, like those built using
the Applicative
class, are rather inflexible: the structure of the computation
is fixed at the outset, and there is no ability to choose between
alternate execution paths based on intermediate results.
The ArrowChoice
class provides exactly such an ability:
class Arrow arr => ArrowChoice arr where
left :: (b `arr` c) > (Either b d `arr` Either c d)
right :: (b `arr` c) > (Either d b `arr` Either d c)
(+++) :: (b `arr` c) > (b' `arr` c') > (Either b b' `arr` Either c c')
() :: (b `arr` d) > (c `arr` d) > (Either b c `arr` d)
A comparison of ArrowChoice
to Arrow
will reveal a striking
parallel between left
, right
, (+++)
, ()
and first
,
second
, (***)
, (&&&)
, respectively. Indeed, they are dual:
first
, second
, (***)
, and (&&&)
all operate on product types
(tuples), and left
, right
, (+++)
, and ()
are the
corresponding operations on sum types. In general, these operations
create arrows whose inputs are tagged with Left
or Right
, and can
choose how to act based on these tags.
 If
g
is an arrow fromb
toc
, thenleft g
is an arrow fromEither b d
toEither c d
. On inputs tagged withLeft
, theleft g
arrow has the behavior ofg
; on inputs tagged withRight
, it behaves as the identity.
 The
right
function, of course, is the mirror image ofleft
. The arrowright g
has the behavior ofg
on inputs tagged withRight
.
 The
(+++)
operator performs “multiplexing”:g +++ h
behaves asg
on inputs tagged withLeft
, and ash
on inputs tagged withRight
. The tags are preserved. The(+++)
operator is the sum (hence+
) of two arrows, just as(***)
is the product.
 The
()
operator is “merge” or “fanin”: the arrowg  h
behaves asg
on inputs tagged withLeft
, andh
on inputs tagged withRight
, but the tags are discarded (hence,g
andh
must have the same output type). The mnemonic is thatg  h
performs eitherg
orh
on its input.
The ArrowChoice
class allows computations to choose among a finite number of execution paths, based on intermediate results. The possible
execution paths must be known in advance, and explicitly assembled with (+++)
or ()
. However, sometimes more flexibility is
needed: we would like to be able to compute an arrow from intermediate results, and use this computed arrow to continue the computation. This is the power given to us by ArrowApply
.
ArrowApply
The ArrowApply
type class is:
class Arrow arr => ArrowApply arr where
app :: (b `arr` c, b) `arr` c
If we have computed an arrow as the output of some previous
computation, then app
allows us to apply that arrow to an input,
producing its output as the output of app
. As an exercise, the
reader may wish to use app
to implement an alternative “curried”
version, app2 :: b `arr` ((b `arr` c) `arr` c)
.
This notion of being able to compute a new computation
may sound familiar:
this is exactly what the monadic bind operator (>>=)
does. It
should not particularly come as a surprise that ArrowApply
and
Monad
are exactly equivalent in expressive power. In particular,
Kleisli m
can be made an instance of ArrowApply
, and any instance
of ArrowApply
can be made a Monad
(via the newtype
wrapper
ArrowMonad
). As an exercise, the reader may wish to try
implementing these instances:
instance Monad m => ArrowApply (Kleisli m) where
app =  exercise
newtype ArrowApply a => ArrowMonad a b = ArrowMonad (a () b)
instance ArrowApply a => Monad (ArrowMonad a) where
return =  exercise
(ArrowMonad a) >>= k =  exercise
ArrowLoop
The ArrowLoop
type class is:
class Arrow a => ArrowLoop a where
loop :: a (b, d) (c, d) > a b c
trace :: ((b,d) > (c,d)) > b > c
trace f b = let (c,d) = f (b,d) in c
It describes arrows that can use recursion to compute results, and is
used to desugar the rec
construct in arrow notation (described
below).
Taken by itself, the type of the loop
method does not seem to tell
us much. Its intention, however, is a generalization of the trace
function which is also shown. The d
component of the first arrow’s
output is fed back in as its own input. In other words, the arrow
loop g
is obtained by recursively “fixing” the second component of
the input to g
.
It can be a bit difficult to grok what the trace
function is doing.
How can d
appear on the left and right sides of the let
? Well,
this is Haskell’s laziness at work. There is not space here for a
full explanation; the interested reader is encouraged to study the
standard fix
function, and to read Paterson’s arrow tutorial.
Arrow notation
Programming directly with the arrow combinators can be painful,
especially when writing complex computations which need to retain
simultaneous reference to a number of intermediate results. With
nothing but the arrow combinators, such intermediate results must be
kept in nested tuples, and it is up to the programmer to remember
which intermediate results are in which components, and to swap,
reassociate, and generally mangle tuples as necessary. This problem
is solved by the special arrow notation supported by GHC, similar to
do
notation for monads, that allows names to be assigned to
intermediate results while building up arrow computations. An example
arrow implemented using arrow notation, taken from
Paterson, is:
class ArrowLoop arr => ArrowCircuit arr where
delay :: b > (b `arr` b)
counter :: ArrowCircuit arr => Bool `arr` Int
counter = proc reset > do
rec output < idA < if reset then 0 else next
next < delay 0 < output + 1
idA < output
This arrow is intended to represent a recursively defined counter circuit with a reset line.
There is not space here for a full explanation of arrow notation; the interested reader should consult Paterson’s paper introducing the notation, or his later tutorial which presents a simplified version.
Further reading
An excellent starting place for the student of arrows is the arrows web page, which contains an introduction and many references. Some key papers on arrows include Hughes’ original paper introducing arrows, Generalising monads to arrows, and Paterson’s paper on arrow notation.
Both Hughes and Paterson later wrote accessible tutorials intended for a broader audience: Paterson: Programming with Arrows and Hughes: Programming with Arrows.
Although Hughes’ goal in defining the Arrow
class was to
generalize Monad
s, and it has been said that Arrow
lies “between
Applicative
and Monad
” in power, they are not directly
comparable. The precise relationship remained in some confusion until
analyzed by Lindley, Wadler, and Yallop, who
also invented a new calculus of arrows, based on the lambda calculus,
which considerably simplifies the presentation of the arrow laws
(see The arrow calculus). There is also a precise technical sense in which Arrow
can be seen as the intersection of Applicative
and Category
.
Some examples of Arrow
s include Yampa, the
Haskell XML Toolkit, and the functional GUI library Grapefruit.
Some extensions to arrows have been explored; for example, the
BiArrow
s of Alimarine et al. ("There and Back Again: Arrows for Invertible Programming"), for twoway instead of oneway
computation.
The Haskell wiki has links to many additional research papers relating to Arrow
s.
Comonad
The final type class we will examine is Comonad
. The Comonad
class
is the categorical dual of Monad
; that is, Comonad
is like Monad
but with all the function arrows flipped. It is not actually in the
standard Haskell libraries, but it has seen some interesting uses
recently, so we include it here for completeness.
Definition
The Comonad
type class, defined in the Control.Comonad
module of
the comonad library, is:
class Functor w => Comonad w where
extract :: w a > a
duplicate :: w a > w (w a)
duplicate = extend id
extend :: (w a > b) > w a > w b
extend f = fmap f . duplicate
As you can see, extract
is the dual of return
, duplicate
is the dual of join
, and extend
is the dual of (=<<)
. The definition of Comonad
is a bit redundant, giving the programmer the choice on whether extend or duplicate are implemented; the other operation then has a default implementation.
A prototypical example of a Comonad
instance is:
 Infinite lazy streams
data Stream a = Cons a (Stream a)
 'duplicate' is like the list function 'tails'
 'extend' computes a new Stream from an old, where the element
 at position n is computed as a function of everything from
 position n onwards in the old Stream
instance Comonad Stream where
extract (Cons x _) = x
duplicate s@(Cons x xs) = Cons s (duplicate xs)
extend g s@(Cons x xs) = Cons (g s) (extend g xs)
 = fmap g (duplicate s)
Further reading
Dan Piponi explains in a blog post what cellular automata have to do with comonads. In another blog post, Conal Elliott has examined a comonadic formulation of functional reactive programming. Sterling Clover’s blog post Comonads in everyday life explains the relationship between comonads and zippers, and how comonads can be used to design a menu system for a web site.
Uustalu and Vene have a number of papers exploring ideas related to comonads and functional programming:
 Comonadic Notions of Computation
 The dual of substitution is redecoration (Also available as ps.gz.)
 Recursive coalgebras from comonads
 Recursion schemes from comonads
 The Essence of Dataflow Programming.
Gabriel Gonzalez's Comonads are objects points out similarities between comonads and objectoriented programming.
The comonadtransformers package contains comonad transformers.
Acknowledgements
A special thanks to all of those who taught me about standard Haskell type classes and helped me develop good intuition for them, particularly Jules Bean (quicksilver), Derek Elkins (ddarius), Conal Elliott (conal), Cale Gibbard (Cale), David House, Dan Piponi (sigfpe), and Kevin Reid (kpreid).
I also thank the many people who provided a mountain of helpful feedback and suggestions on a first draft of the Typeclassopedia: David Amos, Kevin Ballard, Reid Barton, Doug Beardsley, Joachim Breitner, Andrew Cave, David Christiansen, Gregory Collins, Mark Jason Dominus, Conal Elliott, Yitz Gale, George Giorgidze, Steven Grady, Travis Hartwell, Steve Hicks, Philip Hölzenspies, Edward Kmett, Eric Kow, Serge Le Huitouze, Felipe Lessa, Stefan Ljungstrand, Eric Macaulay, Rob MacAulay, Simon Meier, Eric Mertens, Tim Newsham, Russell O’Connor, Conrad Parker, Walt RorieBaety, Colin Ross, Tom Schrijvers, Aditya Siram, C. Smith, Martijn van Steenbergen, Joe Thornber, Jared Updike, Rob Vollmert, Andrew Wagner, Louis Wasserman, and Ashley Yakeley, as well as a few only known to me by their IRC nicks: b_jonas, maltem, tehgeekmeister, and ziman. I have undoubtedly omitted a few inadvertently, which in no way diminishes my gratitude.
Finally, I would like to thank Wouter Swierstra for his fantastic work editing the Monad.Reader, and my wife Joyia for her patience during the process of writing the Typeclassopedia.
About the author
Brent Yorgey (blog, homepage) is (as of November 2011) a fourthyear Ph.D. student in the programming languages group at the University of Pennsylvania. He enjoys teaching, creating EDSLs, playing Bach fugues, musing upon category theory, and cooking tasty lambdatreats for the denizens of #haskell.
Colophon
The Typeclassopedia was written by Brent Yorgey and initially published in March 2009. Painstakingly converted to wiki syntax by User:Geheimdienst in November 2011, after asking Brent’s permission.
If something like this TeX to wiki syntax conversion ever needs to be done again, here are some vim commands that helped:
 %s/\\section{\([^}]*\)}/=\1=/gc
 %s/\\subsection{\([^}]*\)}/==\1==/gc
 %s/^ *\\item /\r* /gc
 %s//—/gc
 %s/\$\([^$]*\)\$/<math>\1\\ <\/math>/gc Appending “\ ” forces images to be rendered. Otherwise, Mediawiki would go back and forth between one font for short <math> tags, and another more TeXlike font for longer tags (containing more than a few characters)""
 %s/\([^]*\)/<code>\1<\/code>/gc
 %s/\\dots/.../gc
 %s/^\\label{.*$//gc
 %s/\\emph{\([^}]*\)}/''\1''/gc
 %s/\\term{\([^}]*\)}/''\1''/gc
The biggest issue was taking the academicpaperstyle citations and turning them into hyperlinks with an appropriate title and an appropriate target. In most cases there was an obvious thing to do (e.g. online PDFs of the cited papers or CiteSeer entries). Sometimes, however, it’s less clear and you might want to check the original Typeclassopedia PDF with the original bibliography file.
To get all the citations into the main text, I first tried processing the source with TeX or Lyx. This didn’t work due to missing unfindable packages, syntax errors, and my general ineptitude with TeX.
I then went for the next best solution, which seemed to be extracting all instances of “\cite{something}” from the source and in that order pulling the referenced entries from the .bib file. This way you can go through the source file and sortedreferences file in parallel, copying over what you need, without searching back and forth in the .bib file. I used:
 egrep o "\cite\{[^\}]*\}" ~/typeclassopedia.lhs  cut c 6  tr "," "\n"  tr d "}" > /tmp/citations
 for i in $(cat /tmp/citations); do grep A99 "$i" ~/typeclassopedia.bibegrep B99 '^\}$' m1 ; done > ~/typeclassorefssorted