Difference between revisions of "Typeclassopedia"
Geheimdienst (talk  contribs) m (→do notation) 
KetilMalde (talk  contribs) m (fix line breaks in bullet points (Arrow intuition)) 

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build intuition. 
build intuition. 

−  * The <code>arr</code> function takes any function <code>b > c</code> and turns it into a 

+  * The <code>arr</code> function takes any function <code>b > c</code> and turns it into a generalized arrow <code>b ~> c</code>. The <code>arr</code> 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 arrow <code>arr g</code> is “pure” in the sense that it only computes <code>g</code> and has no “effects” (whatever that might mean for any particular arrow type). 

−  generalized arrow <code>b ~> c</code>. The <code>arr</code> 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 arrow <code>arr g</code> is 

−  “pure” in the sense that it only computes <code>g</code> and has no 

−  “effects” (whatever that might mean for any particular arrow type). 

−  * The <code>first</code> method turns any arrow from <code>b</code> to <code>c</code> into an arrow 

+  * The <code>first</code> method turns any arrow from <code>b</code> to code>c</code> into an arrow from <code>(b,d)</code> to <code>(c,d)</code>. The idea is that <code>first g</code> uses code>g</code> to process the first element of a tuple, and lets the second element pass through unchanged. For the function instance of <code>Arrow</code>, of course, <code>first g (x,y) = (g x, y)</code>. 

−  from <code>(b,d)</code> to <code>(c,d)</code>. The idea is that <code>first g</code> uses <code>g</code> to 

−  process the first element of a tuple, and lets the second element pass 

−  through unchanged. For the function instance of <code>Arrow</code>, of course, 

−  <code>first g (x,y) = (g x, y)</code>. 

−  * The <code>second</code> function is similar to <code>first</code>, but with the elements of the 
+  * The <code>second</code> function is similar to <code>first</code>, but with the elements of the tuples swapped. Indeed, it can be defined in terms of <code>first</code> using an auxiliary function <code>swap</code>, defined by <code>swap (x,y) = (y,x)</code>. 
−  tuples swapped. Indeed, it can be defined in terms of <code>first</code> using 

−  an auxiliary function <code>swap</code>, defined by <code>swap (x,y) = (y,x)</code>. 

−  * The <code>(***)</code> operator is “parallel composition” of arrows: it takes two 

+  * The <code>(***)</code> 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 that <code>g *** h</code> is the ''product'' (hence <code>*</code>) of <code>g</code> and <code>h</code>. For the function instance of <code>Arrow</code>, we define <code>(g *** h) (x,y) = (g x, h y)</code>. The default implementation of <code>(***)</code> is in terms of <code>first</code>, <code>second</code>, and sequential arrow composition <code>(>>>)</code>. The reader may also wish to think about how to implement <code>first</code> and <code>second</code> in terms of <code>(***)</code>. 

−  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 that <code>g *** h</code> is the ''product'' (hence <code>*</code>) of <code>g</code> and 

−  <code>h</code>. For the function instance of <code>Arrow</code>, 

−  we define <code>(g *** h) (x,y) = (g x, h y)</code>. The default implementation of 

−  <code>(***)</code> is in terms of <code>first</code>, <code>second</code>, and sequential arrow 

−  composition <code>(>>>)</code>. The reader may also wish to think about how to 

−  implement <code>first</code> and <code>second</code> in terms of <code>(***)</code>. 

−  * The <code>(&&&)</code> operator is “fanout composition” of arrows: it takes two arrows 

+  * The <code>(&&&)</code> operator is “fanout composition” of arrows: it takes two arrows <code>g</code> and <code>h</code> and makes them into a new arrow <code>g &&& h</code> which supplies its input as the input to both <code>g</code> and <code>h</code>, returning their results as a tuple. The mnemonic is that <code>g &&& h</code> performs both <code>g</code> ''and'' <code>h</code> (hence <code>&</code>) on its input. For functions, we define <code>(g &&& h) x = (g x, h x)</code>. 

−  <code>g</code> and <code>h</code> and makes them into a new arrow <code>g &&& h</code> which supplies 

−  its input as the input to both <code>g</code> and <code>h</code>, returning their results 

−  as a tuple. The mnemonic is that <code>g &&& h</code> performs both <code>g</code> 

−  ''and'' <code>h</code> (hence <code>&</code>) on its input. For functions, we define <code>(g &&& h) x = (g x, h x)</code>. 

==Instances== 
==Instances== 
Revision as of 07:55, 26 November 2011
 By Brent Yorgey, byorgey@cis.upenn.edu
 As published 12 March 2009, issue 13 of the Monad.Reader, with tiny November 2011 updates by Geheimdienst
 Alternate formats: PDF / tex source / bibliography
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 article 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.
Contents
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 article is to set you on the road to gaining such intuition. However—
 There is no royal road to Haskell. —Euclid
∗ See Brent Yorgey’s Abstraction, intuition, and the “monad tutorial fallacy” This article 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:
∗ When Typeclassopedia was originally written, Pointed
and Comonad
were in the categoryextras library. It has since been deprecated and they have moved to the pointed package and the comonad package. —Geheimdienst, Nov 2011
 Solid arrows point from the general to the specific; that is, if there is an arrow from Foo to Bar it means that every Bar is (or should be, or can be made into) a Foo.
 Dotted arrows indicate some other sort of relationship.

Monad
andArrowApply
are equivalent. 
Pointed
andComonad
are greyed out since they are not actually (yet) in the standard Haskell libraries ∗.
One more note before we begin. I’ve seen “type class” written as one word, “typeclass,” but let’s settle this once and for all: the correct spelling uses two words (the title of this article notwithstanding), as evidenced by, for example, the Haskell 98 Revised 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.
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
. 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
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 tutorial, I will use f
only to represent Functor
s, and g
or h
to 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.

((>) e)
, the type of functions which take a value of typee
as a parameter, is aFunctor
. It would be clearer to write it as(e >)
, by analogy with an operator section like(1 +)
, but that syntax is not allowed. However, you can certainly think of it as(e >)
. 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
,Sequence
, andStream
) 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
.
A good exercise is to implement Functor
instances for Either e
, ((,) e)
, and ((>) e)
.
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.
Intuition
There are two fundamental ways to think about fmap
. The first has already been touched on: 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.
Pointed
∗ The Pointed
type class lives in the pointed library, moved from the categoryextras library. The point
function was originally named pure
.
Edward Kmett, the author of categoryextras, pointed, and many related packages, has since moved his focus to semigroupoids and semigroups. He finds them more interesting and useful, and considers Pointed
to be historical now (he still provides the pointed package only because “people were whinging”). Nevertheless, Pointed
has kept its value for explaining, and its place in Typeclassopedia. —Geheimdienst, Nov 2011
The Pointed
type class represents pointed functors. It is not actually a type class in the standard libraries ∗. But it could be, and it’s useful in understanding a few other type classes, notably Applicative
and Monad
, so let’s pretend for a minute.
Given a Functor
, the Pointed
class represents the additional ability to put a value into a “default context.” Often, this corresponds to creating a container with exactly one element, but it is more general than that. The type class declaration for Pointed
is:
class Functor f => Pointed f where
point :: a > f a  aka pure, singleton, return, unit
Most of the standard Functor
instances could also be instances of Pointed
—for example, the Maybe
instance of Pointed
is point = Just
; there are many possible implementations for lists, the most natural of which is point x = [x]
; for ((>) e)
it is ... well, I’ll let you work it out. (Just follow the types!)
One example of a Functor
which is not Pointed
is ((,) e)
. If you try implementing point :: a > (e,a)
you will quickly see why: since the type e
is completely arbitrary, there is no way to generate a value of type e
out of thin air! However, as we will see, ((,) e)
can be made Pointed
if we place an additional restriction on e
which allows us to generate a default value of type e
(the most common solution is to make e
an instance of Monoid
).
∗ For those interested in category theory, this law states precisely that point
is a natural transformation from the identity functor to f
. The Pointed
class has only one law ∗:
fmap g . point = point . g
∗ ... modulo ⊥, seq
, and assuming a lawful Functor
instance.
However, you need not worry about it: this law is actually a socalled “free theorem” guaranteed by parametricity (see Wadler’s Theorems for free!); it’s impossible to write an instance of Pointed
which does not satisfy it ∗.
Applicative
A somewhat newer addition to the pantheon of standard Haskell type classes, applicative functors represent an abstraction lying exactly in between Functor
and Monad
, 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
The Applicative
class adds a single capability to Pointed
functors. 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 another context. Applicative
gives us just such a tool. Here is the type class declaration for Applicative
, as defined in Control.Applicative
:
class Functor f => Applicative f where
pure :: a > f a  aka point
(<*>) :: 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 signature of (<*>)
. The best way of thinking about it comes from noting that the type of (<*>)
is similar to the type of ($)
∗, but with everything enclosed in an f
. In other words, (<*>)
is just function application within a computational context. The type of (<*>)
is also very similar to the type of fmap
; the only difference is that the first parameter is f (a > b)
, a function in a context, instead of a “normal” function (a > b)
.
Of course, pure
looks rather familiar. It is the point
function from the Pointed
type class. If we actually had it in the standard library, and pure
appearing under the other name didn’t bother you, then Applicative
could instead be defined as:
class Pointed f => Applicative' f where
(<*>) :: f (a > b) > f a > f b
Laws
∗ See haddock for Applicative and Applicative programming with effects
There are several laws that Applicative
instances should satisfy ∗, but only one is crucial to developing intuition, because it specifies how Applicative
should relate to Functor
(the other four mostly specify the exact sense in which pure
deserves its name). This law is:
fmap g x = pure g <*> x
It says that mapping a pure function g
over a context x
is the same as first injecting g
into a context with pure
, and then applying it to x
with (<*>)
. In other words, we can decompose fmap
into two more atomic operations: injection into a context, and application within a context. The Control.Applicative
module also defines (<$>)
as a synonym for fmap
, so the above law can also be expressed as:
g <$> x = pure g <*> x
.
Instances
Most of the standard types which are instances of Functor
are also instances of Applicative
.
Maybe
can easily be made an instance of Applicative
; writing such an instance is left as an exercise for the reader.
The list type constructor []
can actually be made an instance of Applicative
in two ways; essentially, it comes down to whether we want to think of lists as ordered collections of elements, or as contexts representing multiple results of a nondeterministic computation (see Wadler’s How to replace failure by a list of successes).
Let’s first consider the collection point of view. Since there can only be one instance of a given type class for any particular type, one or both of the list instances of Applicative
need to be defined for a newtype
wrapper; as it happens, the nondeterministic computation instance is the default, and the collection instance is defined in terms of a newtype
called ZipList
. This instance is:
newtype ZipList a = ZipList { getZipList :: [a] }
instance Applicative ZipList where
pure = undefined  exercise
(ZipList gs) <*> (ZipList xs) = ZipList (zipWith ($) 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, ($)
; hence the name ZipList
. As an exercise, determine the correct definition of pure
—there is only one implementation that satisfies the law (see section “Laws”).
The other Applicative
instance for lists, based on the nondeterministic computation point of view, is:
instance Applicative [] where
pure x = [x]
gs <*> xs = [ g x  g < gs, x < xs ]
Instead of applying functions to inputs pairwise, we apply each function to all the inputs in turn, and collect all the results in a list.
Now we can write nondeterministic computations in a natural style. To add the numbers 3
and 4
deterministically, we can of course write (+) 3 4
. But suppose instead of 3
we have a nondeterministic computation that might result in 2
, 3
, or 4
; then we can write
pure (+) <*> [2,3,4] <*> pure 4
or, more idiomatically,
(+) <$> [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: wheng <$> m1 <*> m2 <*> m3
is executed, the effects from themi
’s happen in order from left to right.

((,) 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.
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.
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 superceded by Pushpull functional reactive programming. —Geheimdienst, Nov 2011
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.
Monad
It’s a safe bet that if you’re reading this article, 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 various 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
.
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 a pointed functor (indeed, 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 Pointed
or 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 monad transformer library (mtl).
 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; I leave its implementation as an exercise. Follow the types!
 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. It is worth trying to write aMonad
instance for((>) e)
yourself.
 The
Control.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.
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. 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.
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.
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.
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, monads in category theory are defined in terms of return
, fmap
, and join
(often called , , and in the mathematical literature). Haskell uses the equivalent formulation in terms of (>>=)
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
.) An excellent exercise is to implement (>>=)
in terms of fmap
and join
, and to implement join
in terms of (>>=)
.
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.
∗ Still, it is unclear how this "bug" should be fixed. Making Monad
require a Functor
instance has some drawbacks, as mentioned in this 2011 mailinglist discussion. —Geheimdienst

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 it is a bug (in a social rather than technical sense) for any type to be an instance ofMonad
without also being an 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 also 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 the IfElse 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.
 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.
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,” but that’s probably not the canonical pronunciation ...
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
. (Note that return
is such a function.) (>=>)
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 ∗. Working out the equivalence between these two formulations, given the definition g >=> h = \x > g x >>= h
, is left as an exercise.
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.
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.
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.
The monad transformer library mtl provides a number of monad transformers, such as StateT
, ReaderT
, ErrorT
(haddock), and (soon) MaybeT
, which can be applied to other monads to produce a new monad with the effects of both. 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
. These 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; 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”; for example, 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.)
All monad transformers should implement the MonadTrans
type class, defined in Control.Monad.Trans
:
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
. As an exercise, you may wish to work out t
’s kind, which is rather more interesting than most of the kinds we’ve seen up to this point.) However, you should only have to think about MonadTrans
when defining your own monad transformers, not when using predefined ones.
∗ 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 such as MonadState
, which provides statespecific methods like 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 ∗.
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.
MonadFix
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 recursion. This is supported in GHC and Hugs by a special “recursive do” notation, mdo
. For more information, see Levent Erkök’s thesis, Value Recursion in Monadic Computations.
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.
Much of the monad transformer library mtl, including the Reader
, Writer
, State
, and other monads, as well as the monad transformer framework itself, was inspired by Mark Jones’s classic paper Functional Programming with Overloading and HigherOrder Polymorphism. It’s still very much worth a read—and highly readable—after almost fifteen years.
∗ {{{1}}}
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, and Tim Newsham’s What’s a Monad?. There are also many blog articles which have been written on various aspects of monads; a collection of links can be found under Blog articles/Monads.
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 [[Do_notation_considered_harmfulconsider 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’s 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
. There is also an alternative way to compose monads, using coproducts, as described by Lüth and Ghani, although this method has not (yet?) seen widespread use.
Links to many more research papers related to monads can be found under Research papers/Monads and arrows.
Monoid
A monoid 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 ), and there must be some element of which is the identity with respect to . (If you are familiar with group theory, a monoid is like a group without the requirement that inverses exist.) For example, the natural numbers under addition form a monoid: the sum of any two natural numbers is a natural number; for any natural numbers , , and ; and zero is the additive identity. The integers under multiplication also form a monoid, as do natural numbers under , Boolean values under conjunction and disjunction, lists under concatenation, functions from a set to itself under composition ... Monoids show up all over the place, once you know to look for them.
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).
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 lists x
, y
, and z
, 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
two newtype
wrappers, Sum
and Product
, with appropriate
Monoid
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]
and product [1..5]
. Nevertheless, these instances
are useful in more generalized settings, as we will see in the
section Foldable
.

Any
andAll
arenewtype
wrappers providingMonoid
instances for Bool
(under disjunction and conjunction,
respectively).
 There are three instances for
Maybe
: a basic instance which
lifts a Monoid
instance for a
to an instance for Maybe a
, and
two newtype
wrappers First
and Last
for which mappend
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 for a
can be lifted to an instance for Maybe a
. There
are also tuple instances: if a
and b
are instances of Monoid
,
then so is (a,b)
, using the monoid operations for a
and b
in
the obvious pairwise manner. Finally, if a
is a Monoid
, then so
is the function type e > a
for any e
; in particular,
g `mappend` h
is the function which applies both g
and h
to
its argument and then combines the result using the underlying
Monoid
instance for a
. This can be quite useful and
elegant (see example).
 The type
Ordering = LT  EQ  GT
is aMonoid
, defined in
such a way that
mconcat (zipWith compare xs ys)
computes the
lexicographic ordering of xs
and ys
. In particular,
mempty = EQ
, and mappend
evaluates to its leftmost nonEQ
argument (or EQ
if both arguments are EQ
). This can be used
together with the function instance of Monoid
to do some clever
things
(example).
 There are also
Monoid
instances for several standard data
structures in the containers library (haddock),
including Map
, Set
, and Sequence
.
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.
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 (~>) => ArrowZero (~>) where
zeroArrow :: b ~> c
class ArrowZero (~>) => ArrowPlus (~>) where
(<+>) :: (b ~> c) > (b ~> c) > (b ~> 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, [http://blog.sigfpe.com/2009/01/haskellmonoidsandtheiruses.html Haskell Monoids and their
Uses]. This was quickly followed by
Heinrich Apfelmus’s 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 [http://www.reddit.com/r/programming/comments/7cf4r/monoids_in_my_programming_language/c06adnx building
elegant list sorting combinators],
collecting unstructured information,
and a brilliant series of posts by ChungChieh Shan and Dylan Thurston
using Monoid
s to [http://conway.rutgers.edu/~ccshan/wiki/blog/posts/WordNumbers1/ 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 ++ f k ++ foldMap f r
where (++) = mappend
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.
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 reader may enjoy
coming up with elegant implementations of these functions using fold
or foldMap
and appropriate Monoid
instances.
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
. 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.
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? (Answer: yes, in two ways. Figuring out what they are, and why, is left as an exercise. A much more challenging question is whether a list of trees can be turned into a tree of lists.)
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’s paper for more details.
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. A good exercise is to implement fmap
and foldMap
using
only the Traversable
methods; the implementations are surprisingly
elegant. The Traversable
module provides these
implementations as fmapDefault
and foldMapDefault
.
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
.
Further reading
The Traversable
class also had its genesis in [http://www.soi.city.ac.uk/~ross/papers/Applicative.html 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.
Category
Category
is another fairly new addition to the Haskell standard
libraries; you may or may not have it installed depending on the
version of your base
package. It generalizes the notion of
function composition to general “morphisms.”
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 constructor (~>)
, much like the infix function type
constructor (>)
. This syntax is not part of Haskell 98.
The second definition shown is the one used in the standard libraries.
For the remainder of the article, I will use the infix type
constructor (~>)
for Category
as well as Arrow
.
class Category (~>) where
id :: a ~> a
(.) :: (b ~> c) > (a ~> b) > (a ~> 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’s 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 (~>)
is an instance of Arrow
, a value of
type b ~> 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 (~>) => Arrow (~>) where
arr :: (b > c) > (b ~> c)
first :: (b ~> c) > ((b, d) ~> (c, d))
second :: (b ~> c) > ((d, b) ~> (d, c))
(***) :: (b ~> c) > (b' ~> c') > ((b, b') ~> (c, c'))
(&&&) :: (b ~> c) > (b ~> c') > (b ~> (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 ~> c
and h :: c ~> d
, we can form their
composition g >>> h :: b ~> 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 ~> 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
to code>c</code> into an arrow from(b,d)
to(c,d)
. The idea is thatfirst g
uses code>g</code> 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 (~>) => ArrowChoice (~>) where
left :: (b ~> c) > (Either b d ~> Either c d)
right :: (b ~> c) > (Either d b ~> Either d c)
(+++) :: (b ~> c) > (b' ~> c') > (Either b b' ~> Either c c')
() :: (b ~> d) > (c ~> d) > (Either b c ~> 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
from Either b d
to Either c d
. On inputs tagged with Left
,
the left g
arrow has the behavior of g
; on inputs tagged with Right
, it
behaves as the identity.
 The
right
function, of course, is the mirror image ofleft
. The arrowright g
has the behavior of g
on inputs tagged with Right
.
 The
(+++)
operator performs “multiplexing”:g +++ h
behaves asg
on inputs tagged with Left
, and as h
on inputs tagged with
Right
. 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 as g
on inputs tagged with Left
, and h
on inputs
tagged with Right
, but the tags are discarded (hence, g
and h
must have the same output type). The mnemonic is that g  h
performs either g
or h
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 (~>) => ArrowApply (~>) where
app :: (b ~> c, b) ~> 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 ~> ((b ~> c) ~> 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 (~>) => ArrowCircuit (~>) where
delay :: b > (b ~> b)
counter :: ArrowCircuit (~>) => Bool ~> 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 [http://www.soi.city.ac.uk/~ross/papers/notation.html Paterson’s paper introducing the notation], or his later [http://www.soi.city.ac.uk/~ross/papers/fop.html 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’s 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’s 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).
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., 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 categoryextras library, is:
class Functor f => Copointed f where
extract :: f a > a
class Copointed w => Comonad w where
duplicate :: w a > w (w a)
extend :: (w a > b) > w a > w b
As you can see, extract
is the dual of return
, duplicate
is the
dual of join
, and extend
is the dual of (>>=)
(although its
arguments are in a different order). The definition
of Comonad
is a bit redundant (after all, the Monad
class does not
need join
), but this is so that a Comonad
can be defined by fmap
,
extract
, and either duplicate
or extend
. Each has a
default implementation in terms of the other.
A prototypical example of a Comonad
instance is:
 Infinite lazy streams
data Stream a = Cons a (Stream a)
instance Functor Stream where
fmap g (Cons x xs) = Cons (g x) (fmap g xs)
instance Copointed Stream where
extract (Cons x _) = x
 '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
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 [http://blog.sigfpe.com/2006/12/evaluatingcellularautomatais.html cellular automata have to do with comonads]. In another blog post, Conal Elliott has examined [http://conal.net/blog/posts/functionalinteractivebehavior/ 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
 Recursive coalgebras from comonads
 Recursion schemes from comonads
 The Essence of Dataflow Programming.
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 this article: 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 a firstyear 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 initally 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