An example: Maybe

One example of a monad is the Maybe type. Undefined null results are one particular pain point that many procedural languages don't provide specific tools for dealing with, requiring use of the null object pattern or checks to test for invalid values at each operation to handle undefined values. This causes bugs and makes it harder to build robust software that gracefully handles errors. The Maybe type forces the programmer to deal with these potentially undefined results by explicitly defining the two states of a result: Just ⌑result⌑, or Nothing. For example the programmer might be constructing a parser, which is to return an intermediate result, or else signal a condition which the parser has detected, and which programmer must also handle. With just a little extra functional spice on top, this Maybe type transforms into a fully-featured monad.^{[lower-alpha 2]: 12.3 pages 148–151}

In most languages, the Maybe monad is also known as an option type, which is just a type that marks whether or not it contains a value. Typically they are expressed as some kind of enumerated type. In this Rust example we will call it Maybe<T> and variants of this type can either be a value of generic type T, or the empty variant: Nothing.

// The <T> represents a generic type "T"
enum Maybe<T> {
    Just(T),
    Nothing,
}

Maybe<T> can also be understood as a "wrapping" type, and this is where its connection to monads comes in. In languages with some form of the Maybe type, there are functions that aid in their use such as composing monadic functions with each other and testing if a Maybe contains a value.

In the following hard-coded example, a Maybe type is used as a result of functions that may fail, in this case the type returns nothing if there is a divide-by-zero.

fn divide(x: Decimal, y: Decimal) -> Maybe<Decimal> {
    if y == 0 { return Nothing }
    else { return Just(x / y) }
}
// divide(1.0, 4.0) -> returns Just(0.25)
// divide(3.0, 0.0) -> returns Nothing

One such way to test whether or not a Maybe contains a value is to use if statements.

let m_x = divide(3.14, 0.0); // see divide function above
// The if statement extracts x from m_x if m_x is the Just variant of Maybe
if let Just(x) = m_x {
    print!("answer: ", x)
} else {
    print!("division failed, divide by zero error...")
}

Other languages may have pattern matching

let result = divide(3.0, 2.0);
match result {
    Just(x) => print!("Answer: ", x),
    Nothing => print!("division failed; we'll get 'em next time."),
}

Monads can compose functions that return Maybe, putting them together. A concrete example might have one function take in several Maybe parameters, and return a single Maybe whose value is Nothing when any of the parameters are Nothing, as in the following:

fn chainable_division(maybe_x: Maybe<Decimal>, maybe_y: Maybe<Decimal>) -> Maybe<Decimal> {
    match (maybe_x, maybe_y) {
        (Just(x), Just(y)) => { // If both inputs are Just, check for division by zero and divide accordingly
            if y == 0 { return Nothing }
            else { return Just(x / y) }
        },
        _ => return Nothing // Otherwise return Nothing
    }
}
chainable_division(chainable_division(Just(2.0), Just(0.0)), Just(1.0)); // inside chainable_division fails, outside chainable_division returns Nothing

Having to rewrite functions to take Maybes in this concrete example requires a lot of boilerplate (look at all those Just expressions!). Instead, we can use something called a bind operator. (also known as "map", "flatmap", or "shove"^[8]: 2205s). This operation takes a monad and a function that returns a monad and runs the function on the inner value of the passed monad, returning the monad from the function.

// Rust example using ".map". maybe_x is passed through 2 functions that return Maybe<Decimal> and Maybe<String> respectively.
// As with normal function composition the inputs and outputs of functions feeding into each other should match wrapped types. (i.e. the add_one function should return a Maybe<Decimal> which then can be unwrapped to a Decimal for the decimal_to_string function)
let maybe_x: Maybe<Decimal> = Just(1.0)
let maybe_result = maybe_x.map(add_one).map(decimal_to_string)

In Haskell, there is an operator bind, or (>>=) that allows for this monadic composition in a more elegant form similar to function composition.^{[lower-alpha 3]: 150–151}

halve :: Int -> Maybe Int
halve x
  | even x = Just (x `div` 2)
  | odd x  = Nothing
 -- This code halves x twice. it evaluates to Nothing if x is not a multiple of 4
halve x >>= halve

With >>= available, chainable_division can be expressed much more succinctly with the help of anonymous functions (i.e. lambdas). Notice in the expression below how the two nested lambdas each operate on the wrapped value in the passed Maybe monad using the bind operator.^{[lower-alpha 4]: 93}

 chainable_division(mx,my) =   mx >>=  ( λx ->   my >>= (λy -> Just (x / y))   )

What has been shown so far is basically a monad, but to be more concise, the following is a strict list of qualities necessary for a monad as defined by the following section.

Monadic Type

A type (Maybe)^{[lower-alpha 2]: 148–151}

Unit operation

A type converter (Just(x))^{[lower-alpha 4]: 93}

Bind operation

A combinator for monadic functions ( >>= or .flatMap())^{[lower-alpha 3]: 150–151}

These are the 3 things necessary to form a monad. Other monads may embody different logical processes, and some may have additional properties, but all of them will have these three similar components.^[1][9]

Definition

The more common definition for a monad in functional programming, used in the above example, is actually based on a Kleisli triple ⟨T, η, μ⟩ rather than category theory's standard definition. The two constructs turn out to be mathematically equivalent, however, so either definition will yield a valid monad. Given any well-defined, basic types T, U, a monad consists of three parts:

• A type constructor M that builds up a monadic type M T^{[lower-alpha 5]}
• A type converter, often called unit or return, that embeds an object x in the monad:
  unit : T → M T^{[lower-alpha 6]}
• A combinator, typically called bind (as in binding a variable) and represented with an infix operator >>= or a method called flatMap, that unwraps a monadic variable, then inserts it into a monadic function/expression, resulting in a new monadic value:
  (>>=) : (M T, T → M U) → M U^{[lower-alpha 7]} so if mx : M T and f : T → M U, then (mx >>= f) : M U

To fully qualify as a monad though, these three parts must also respect a few laws:

• unit is a left-identity for bind:
  unit(x) >>= f ↔ f(x)
• unit is also a right-identity for bind:
  ma >>= unit ↔ ma
• bind is essentially associative:^{[lower-alpha 8]}
  ma >>= λx → (f(x) >>= g) ↔ (ma >>= f) >>= g^[1]

Algebraically, this means any monad both gives rise to a category (called the Kleisli category) and a monoid in the category of functors (from values to computations), with monadic composition as a binary operator in the monoid^[8]: 2450s and unit as identity in the monad.

Usage

The value of the monad pattern goes beyond merely condensing code and providing a link to mathematical reasoning. Whatever language or default programming paradigm a developer uses, following the monad pattern brings many of the benefits of purely functional programming. By reifying a specific kind of computation, a monad not only encapsulates the tedious details of that computational pattern, but it does so in a declarative way, improving the code's clarity. As monadic values explicitly represent not only computed values, but computed effects, a monadic expression can be replaced with its value in referentially transparent positions, much like pure expressions can be, allowing for many techniques and optimizations based on rewriting.^[4]

Typically, programmers will use bind to chain monadic functions into a sequence, which has led some to describe monads as "programmable semicolons", a reference to how many imperative languages use semicolons to separate statements.^[1][5] However, it should be stressed that monads do not actually order computations; even in languages that use them as central features, simpler function composition can arrange steps within a program. A monad's general utility rather lies in simplifying a program's structure and improving separation of concerns through abstraction.^[4][11]

The monad structure can also be seen as a uniquely mathematical and compile time variation on the decorator pattern. Some monads can pass along extra data that is inaccessible to functions, and some even exert finer control over execution, for example only calling a function under certain conditions. Because they let application programmers implement domain logic while offloading boilerplate code onto pre-developed modules, monads can even be considered a tool for aspect-oriented programming.^[12]

One other noteworthy use for monads is isolating side-effects, like input/output or mutable state, in otherwise purely functional code. Even purely functional languages can still implement these "impure" computations without monads, via an intricate mix of function composition and continuation-passing style (CPS) in particular.^[2] With monads though, much of this scaffolding can be abstracted away, essentially by taking each recurring pattern in CPS code and bundling it into a distinct monad.^[4]

If a language does not support monads by default, it is still possible to implement the pattern, often without much difficulty. When translated from category-theory to programming terms, the monad structure is a generic concept and can be defined directly in any language that supports an equivalent feature for bounded polymorphism. A concept's ability to remain agnostic about operational details while working on underlying types is powerful, but the unique features and stringent behavior of monads set them apart from other concepts.^[13]

Verifying the monad laws

Returning to the Maybe example, its components were declared to make up a monad, but no proof was given that it satisfies the monad laws.

This can be rectified by plugging the specifics of Maybe into one side of the general laws, then algebraically building a chain of equalities to reach the other side:

Law 1:  eta(a) >>= f(x)  ⇔  (Just a) >>= f(x)  ⇔  f(a)

Law 2:  ma >>= eta(x)           ⇔  ma

        if ma is (Just a) then
            eta(a)              ⇔ Just a
        else                        or
            Nothing             ⇔ Nothing
        end if

Law 3:  (ma >>= f(x)) >>= g(y)                       ⇔  ma >>= (f(x) >>= g(y))

        if (ma >>= f(x)) is (Just b) then               if ma is (Just a) then
            g(ma >>= f(x))                                (f(x) >>= g(y)) a
        else                                            else
            Nothing                                         Nothing
        end if                                          end if

                ⇔  if ma is (Just a) and f(a) is (Just b) then      
                       (g ∘ f) a
                   else if ma is (Just a) and f(a) is Nothing then
                       Nothing
                   else
                       Nothing
                   end if

Derivation from functors

Though rarer in computer science, one can use category theory directly, which defines a monad as a functor with two additional natural transformations.^{[lower-alpha 10]} So to begin, a structure requires a higher-order function (or "functional") named map to qualify as a functor:

This is not always a major issue, however, especially when a monad is derived from a pre-existing functor, whereupon the monad inherits map automatically. (For historical reasons, this map is instead called fmap in Haskell.)

A monad's first transformation is actually the same unit from the Kleisli triple, but following the hierarchy of structures closely, it turns out unit characterizes an applicative functor, an intermediate structure between a monad and a basic functor. In the applicative context, unit is sometimes referred to as pure but is still the same function. What does differ in this construction is the law unit must satisfy; as bind is not defined, the constraint is given in terms of map instead:

The final leap from applicative functor to monad comes with the second transformation, the join function (in category theory this is a natural transformation usually called μ), which "flattens" nested applications of the monad:

As the characteristic function, join must also satisfy three variations on the monad laws:

Regardless of whether a developer defines a direct monad or a Kleisli triple, the underlying structure will be the same, and the forms can be derived from each other easily:

Another example: List

The List monad naturally demonstrates how deriving a monad from a simpler functor can come in handy. In many languages, a list structure comes pre-defined along with some basic features, so a List type constructor and append operator (represented with ++ for infix notation) are assumed as already given here.

Embedding a plain value in a list is also trivial in most languages:

unit(x)  =  [x]

From here, applying a function iteratively with a list comprehension may seem like an easy choice for bind and converting lists to a full monad. The difficulty with this approach is that bind expects monadic functions, which in this case will output lists themselves; as more functions are applied, layers of nested lists will accumulate, requiring more than a basic comprehension.

However, a procedure to apply any simple function over the whole list, in other words map, is straightforward:

(map φ) xlist  =  [ φ(x1), φ(x2), ..., φ(xn) ]

Now, these two procedures already promote List to an applicative functor. To fully qualify as a monad, only a correct notion of join to flatten repeated structure is needed, but for lists, that just means unwrapping an outer list to append the inner ones that contain values:

join(xlistlist)  =  join([xlist1, xlist2, ..., xlistn])
                 =  xlist1 ++ xlist2 ++ ... ++ xlistn

The resulting monad is not only a list, but one that automatically resizes and condenses itself as functions are applied. bind can now also be derived with just a formula, then used to feed List values through a pipeline of monadic functions:

The List monad can greatly simplify the use of multivalued functions, such as complex roots.^[29]

(xlist >>= f)  =  join ∘ (map f) xlist

One application for this monadic list is representing nondeterministic computation. List can hold results for all execution paths in an algorithm, then condense itself at each step to "forget" which paths led to which results (a sometimes important distinction from deterministic, exhaustive algorithms). Another benefit is that checks can be embedded in the monad; specific paths can be pruned transparently at their first point of failure, with no need to rewrite functions in the pipeline.^[28]

A second situation where List shines is composing multivalued functions. For instance, the nth complex root of a number should yield n distinct complex numbers, but if another mth root is then taken of those results, the final m•n values should be identical to the output of the m•nth root. List completely automates this issue away, condensing the results from each step into a flat, mathematically correct list.^[29]

Syntactic sugar do-notation

Although using bind openly often makes sense, many programmers prefer a syntax that mimics imperative statements (called do-notation in Haskell, perform-notation in OCaml, computation expressions in F#,^[30] and for comprehension in Scala). This is only syntactic sugar that disguises a monadic pipeline as a code block; the compiler will then quietly translate these expressions into underlying functional code.

Translating the add function from the Maybe into Haskell can show this feature in action. A non-monadic version of add in Haskell looks like this:

add mx my =
    case mx of
        Nothing -> Nothing
        Just x  -> case my of
                       Nothing -> Nothing
                       Just y  -> Just (x + y)

In monadic Haskell, return is the standard name for unit, plus lambda expressions must be handled explicitly, but even with these technicalities, the Maybe monad makes for a cleaner definition:

add mx my =
    mx >>= (\x ->
        my >>= (\y ->
            return (x + y)))

With do-notation though, this can be distilled even further into a very intuitive sequence:

add mx my = do
    x <- mx
    y <- my
    return (x + y)

A second example shows how Maybe can be used in an entirely different language: F#. With computation expressions, a "safe division" function that returns None for an undefined operand or division by zero can be written as:

let readNum () =
  let s = Console.ReadLine()
  let succ,v = Int32.TryParse(s)
  if (succ) then Some(v) else None

let secure_div = 
  maybe { 
    let! x = readNum()
    let! y = readNum()
    if (y = 0) 
    then None
    else return (x / y)
  }

At build-time, the compiler will internally "de-sugar" this function into a denser chain of bind calls:

maybe.Delay(fun () ->
  maybe.Bind(readNum(), fun x ->
    maybe.Bind(readNum(), fun y ->
      if (y=0) then None else maybe.Return(x / y))))

For a last example, even the general monad laws themselves can be expressed in do-notation:

do { x <- return v; f x }            ==  do { f v }
do { x <- m; return x }              ==  do { m }
do { y <- do { x <- m; f x }; g y }  ==  do { x <- m; y <- f x; g y }

While convenient, a developer should always remember that this block style is purely syntactic and can be replaced with outwardly monadic (or even non-monadic CPS) expressions. Using bind to express the monadic pipeline can still be clearer in many cases, and some functional programming advocates even argue that since block-style allows novices to carry over habits from imperative programming, it should be avoided by default and only used when obviously superior.^[31][1]

General interface

Every monad needs a specific implementation that meets the monad laws, but other aspects like the relation to other structures or standard idioms within a language are shared by all monads. As a result, a language or library may provide a general Monad interface with function prototypes, subtyping relationships, and other general facts. Besides providing a head-start to development and guaranteeing a new monad inherits features from a supertype (such as functors), checking a monad's design against the interface adds another layer of quality control.

Operators

Monadic code can often be simplified even further through the judicious use of operators. The map functional can be especially helpful since it works on more than just ad-hoc monadic functions; so long as a monadic function should work analogously to a predefined operator, map can be used to instantly "lift" the simpler operator into a monadic one.^{[lower-alpha 11]} With this technique, the definition of add from the Maybe example could be distilled into:

add(mx,my)  =  map (+)

The process could be taken even one step further by defining add not just for Maybe, but for the whole Monad interface. By doing this, any new monad that matches the structure interface and implements its own map will immediately inherit a lifted version of add too. The only change to the function needed is generalizing the type signature:

add : (Monad Number, Monad Number)  →  Monad Number[32]

Another monadic operator that is also useful for analysis is monadic composition (represented as infix >=> here), which allows chaining monadic functions in a more mathematical style:

(f >=> g)(x)  =  f(x) >>= g

With this operator, the monad laws can be written in terms of functions alone, highlighting the correspondence to associativity and existence of an identity:

(unit >=> g)     ↔  g
(f >=> unit)     ↔  f
(f >=> g) >=> h  ↔  f >=> (g >=> h)[1]

In turn, the above shows the meaning of the "do" block in Haskell:

do
 _p <- f(x)
 _q <- g(_p)
 h(_q)          ↔ ( f >=> g >=> h )(x)

Identity monad

The simplest monad is the Identity monad, which just annotates plain values and functions to satisfy the monad laws:

newtype Id T  =  T

unit(x)    =  x
(x >>= f)  =  f(x)

Identity does actually have valid uses though, such as providing a base case for recursive monad transformers. It can also be used to perform basic variable assignment within an imperative-style block.^{[lower-alpha 12]}

Collections

Any collection with a proper append is already a free monoid, but it turns out that List is not the only collection that also has a well-defined join and qualifies as a monad. One can even mutate List into these other monadic collections by simply imposing special properties on append:^{[lower-alpha 13][lower-alpha 14]}

IO monad (Haskell)

As already mentioned, pure code should not have unmanaged side effects, but that does not preclude a program from explicitly describing and managing effects. This idea is central to Haskell's IO monad, where an object of type IO a can be seen as describing an action to be performed in the world, optionally providing information about the world of type a. An action that provides no information about the world has the type IO (), "providing" the dummy value (). When a programmer binds an IO value to a function, the function computes the next action to be performed based on the information about the world provided by the previous action (input from users, files, etc.).^[23] Most significantly, because the value of the IO monad can only be bound to a function that computes another IO monad, the bind function imposes a discipline of a sequence of actions where the result of an action can only be provided to a function that will compute the next action to perform. This means that actions which do not need to be performed never are, and actions that do need to be performed have a well defined sequence, solving the problem of (IO) actions not being referentially transparent.

For example, Haskell has several functions for acting on the wider file system, including one that checks whether a file exists and another that deletes a file. Their two type signatures are:

doesFileExist :: FilePath -> IO Bool
removeFile :: FilePath -> IO ()

The first is interested in whether a given file really exists, and as a result, outputs a Boolean value within the IO monad. The second function, on the other hand, is only concerned with acting on the file system so the IO container it outputs is empty.

IO is not limited just to file I/O though; it even allows for user I/O, and along with imperative syntax sugar, can mimic a typical "Hello, World!" program:

main :: IO ()
main = do
  putStrLn "Hello, world!"
  putStrLn "What is your name, user?"
  name <- getLine
  putStrLn ("Nice to meet you, " ++ name ++ "!")

Desugared, this translates into the following monadic pipeline (>> in Haskell is just a variant of bind for when only monadic effects matter and the underlying result can be discarded):

main :: IO ()
main =
  putStrLn "Hello, world!" >>
  putStrLn "What is your name, user?" >> 
  getLine >>= (\name ->
    putStrLn ("Nice to meet you, " ++ name ++ "!"))

Writer monad (JavaScript)

Another common situation is keeping a log file or otherwise reporting a program's progress. Sometimes, a programmer may want to log even more specific, technical data for later profiling or debugging. The Writer monad can handle these tasks by generating auxiliary output that accumulates step-by-step.

To show how the monad pattern is not restricted to primarily functional languages, this example implements a Writer monad in JavaScript. First, an array (with nested tails) allows constructing the Writer type as a linked list. The underlying output value will live in position 0 of the array, and position 1 will implicitly hold a chain of auxiliary notes:

const writer = value => [value, []];

Defining unit is also very simple:

const unit = value => [value, []];

Only unit is needed to define simple functions that output Writer objects with debugging notes:

const squared = x => [x * x, [`${x} was squared.`]];
const halved = x => [x / 2, [`${x} was halved.`]];

A true monad still requires bind, but for Writer, this amounts simply to concatenating a function's output to the monad's linked list:

const bind = (writer, transform) => {
    const [value, log] = writer;
    const [result, updates] = transform(value);
    return [result, log.concat(updates)];
};

The sample functions can now be chained together using bind, but defining a version of monadic composition (called pipelog here) allows applying these functions even more succinctly:

const pipelog = (writer, ...transforms) =>
    transforms.reduce(bind, writer);

The final result is a clean separation of concerns between stepping through computations and logging them to audit later:

pipelog(unit(4), squared, halved);
// Resulting writer object = [8, ['4 was squared.', '16 was halved.']]

Environment monad

An environment monad (also called a reader monad and a function monad) allows a computation to depend on values from a shared environment. The monad type constructor maps a type T to functions of type E → T, where E is the type of the shared environment. The monad functions are:

$${\displaystyle {\begin{array}{ll}{\text{return}}\colon &T\rightarrow E\rightarrow T=t\mapsto e\mapsto t\\{\text{bind}}\colon &(E\rightarrow T)\rightarrow (T\rightarrow E\rightarrow T')\rightarrow E\rightarrow T'=r\mapsto f\mapsto e\mapsto f\,(r\,e)\,e\end{array}}}$$

The following monadic operations are useful:

$${\displaystyle {\begin{array}{ll}{\text{ask}}\colon &E\rightarrow E={\text{id}}_{E}\\{\text{local}}\colon &(E\rightarrow E)\rightarrow (E\rightarrow T)\rightarrow E\rightarrow T=f\mapsto c\mapsto e\mapsto c\,(f\,e)\end{array}}}$$

The ask operation is used to retrieve the current context, while local executes a computation in a modified subcontext. As in a state monad, computations in the environment monad may be invoked by simply providing an environment value and applying it to an instance of the monad.

Formally, a value in an environment monad is equivalent to a function with an additional, anonymous argument; return and bind are equivalent to the K and S combinators, respectively, in the SKI combinator calculus.

State monads

A state monad allows a programmer to attach state information of any type to a calculation. Given any value type, the corresponding type in the state monad is a function which accepts a state, then outputs a new state (of type s) along with a return value (of type t). This is similar to an environment monad, except that it also returns a new state, and thus allows modeling a mutable environment.

type State s t = s -> (t, s)

Note that this monad takes a type parameter, the type of the state information. The monad operations are defined as follows:

-- "return" produces the given value without changing the state.
return x = \s -> (x, s)
-- "bind" modifies m so that it applies f to its result.
m >>= f = \r -> let (x, s) = m r in (f x) s

Useful state operations include:

get = \s -> (s, s) -- Examine the state at this point in the computation.
put s = \_ -> ((), s) -- Replace the state.
modify f = \s -> ((), f s) -- Update the state

Another operation applies a state monad to a given initial state:

runState :: State s a -> s -> (a, s)
runState t s = t s

do-blocks in a state monad are sequences of operations that can examine and update the state data.

Informally, a state monad of state type S maps the type of return values T into functions of type ${\displaystyle S\rightarrow T\times S}$, where S is the underlying state. The return and bind function are:

${\displaystyle {\begin{array}{ll}{\text{return}}\colon &T\rightarrow S\rightarrow T\times S=t\mapsto s\mapsto (t,s)\\{\text{bind}}\colon &(S\rightarrow T\times S)\rightarrow (T\rightarrow S\rightarrow T'\times S)\rightarrow S\rightarrow T'\times S\ =m\mapsto k\mapsto s\mapsto (k\ t\ s')\quad {\text{where}}\;(t,s')=m\,s\end{array}}}$.

From the category theory point of view, a state monad is derived from the adjunction between the product functor and the exponential functor, which exists in any cartesian closed category by definition.

Continuation monad

A continuation monad^{[lower-alpha 15]} with return type R maps type T into functions of type ${\displaystyle \left(T\rightarrow R\right)\rightarrow R}$. It is used to model continuation-passing style. The return and bind functions are as follows:

${\displaystyle {\begin{array}{ll}{\text{return}}\colon &T\rightarrow \left(T\rightarrow R\right)\rightarrow R=t\mapsto f\mapsto f\,t\\{\text{bind}}\colon &\left(\left(T\rightarrow R\right)\rightarrow R\right)\rightarrow \left(T\rightarrow \left(T'\rightarrow R\right)\rightarrow R\right)\rightarrow \left(T'\rightarrow R\right)\rightarrow R=c\mapsto f\mapsto k\mapsto c\,\left(t\mapsto f\,t\,k\right)\end{array}}}$

The call-with-current-continuation function is defined as follows:

${\displaystyle {\text{call/cc}}\colon \ \left(\left(T\rightarrow \left(T'\rightarrow R\right)\rightarrow R\right)\rightarrow \left(T\rightarrow R\right)\rightarrow R\right)\rightarrow \left(T\rightarrow R\right)\rightarrow R=f\mapsto k\mapsto \left(f\left(t\mapsto x\mapsto k\,t\right)\,k\right)}$

Program logging

The following code is pseudocode. Suppose we have two functions foo and bar, with types

foo : int -> int
bar : int -> int

That is, both functions take in an integer and return another integer. Then we can apply the functions in succession like so:

foo (bar x)

Where the result is the result of foo applied to the result of bar applied to x.

But suppose we are debugging our program, and we would like to add logging messages to foo and bar. So we change the types as so:

foo : int -> int * string
bar : int -> int * string

So that both functions return a tuple, with the result of the application as the integer, and a logging message with information about the applied function and all the previously applied functions as the string.

Unfortunately, this means we can no longer compose foo and bar, as their input type int is not compatible with their output type int * string. And although we can again gain composability by modifying the types of each function to be int * string -> int * string, this would require us to add boilerplate code to each function to extract the integer from the tuple, which would get tedious as the number of such functions increases.

Instead, let us define a helper function to abstract away this boilerplate for us:

bind : int * string -> (int -> int * string) -> int * string

bind takes in an integer and string tuple, then takes in a function (like foo) that maps from an integer to an integer and string tuple. Its output is an integer and string tuple, which is the result of applying the input function to the integer within the input integer and string tuple. In this way, we only need to write boilerplate code to extract the integer from the tuple once, in bind.

Now we have regained some composability. For example:

bind (bind (x,s) bar) foo

Where (x,s) is an integer and string tuple.^{[lower-alpha 16]}

To make the benefits even clearer, let us define an infix operator as an alias for bind:

(>>=) : int * string -> (int -> int * string) -> int * string

So that t >>= f is the same as bind t f.

Then the above example becomes:

((x,s) >>= bar) >>= foo

Finally, it would be nice to not have to write (x, "") every time we wish to create an empty logging message, where "" is the empty string. So let us define a new function:

return : int -> int * string

Which wraps x in the tuple described above.

Now we have a nice pipeline for logging messages:

((return x) >>= bar) >>= foo

That allows us to more easily log the effects of bar and foo on x.

int * string denotes a pseudo-coded monadic value.^{[lower-alpha 16]} bind and return are analogous to the corresponding functions of the same name. In fact, int * string, bind, and return form a monad.

Additive monads

An additive monad is a monad endowed with an additional closed, associative, binary operator mplus and an identity element under mplus, called mzero. The Maybe monad can be considered additive, with Nothing as mzero and a variation on the OR operator as mplus. List is also an additive monad, with the empty list [] acting as mzero and the concatenation operator ++ as mplus.

Intuitively, mzero represents a monadic wrapper with no value from an underlying type, but is also considered a "zero" (rather than a "one") since it acts as an absorber for bind, returning mzero whenever bound to a monadic function. This property is two-sided, and bind will also return mzero when any value is bound to a monadic zero function.

In category-theoretic terms, an additive monad qualifies once as a monoid over monadic functions with bind (as all monads do), and again over monadic values via mplus.^{[33][lower-alpha 17]}

Free monads

Sometimes, the general outline of a monad may be useful, but no simple pattern recommends one monad or another. This is where a free monad comes in; as a free object in the category of monads, it can represent monadic structure without any specific constraints beyond the monad laws themselves. Just as a free monoid concatenates elements without evaluation, a free monad allows chaining computations with markers to satisfy the type system, but otherwise imposes no deeper semantics itself.

For example, by working entirely through the Just and Nothing markers, the Maybe monad is in fact a free monad. The List monad, on the other hand, is not a free monad since it brings extra, specific facts about lists (like append) into its definition. One last example is an abstract free monad:

data Free f a
  = Pure a
  | Free (f (Free f a))

unit :: a -> Free f a
unit x = Pure x

bind :: Functor f => Free f a -> (a -> Free f b) -> Free f b
bind (Pure x) f = f x
bind (Free x) f = Free (fmap (\y -> bind y f) x)

Free monads, however, are not restricted to a linked-list like in this example, and can be built around other structures like trees.

Using free monads intentionally may seem impractical at first, but their formal nature is particularly well-suited for syntactic problems. A free monad can be used to track syntax and type while leaving semantics for later, and has found use in parsers and interpreters as a result.^[34] Others have applied them to more dynamic, operational problems too, such as providing iteratees within a language.^[35]

Comonads

Besides generating monads with extra properties, for any given monad, one can also define a comonad. Conceptually, if monads represent computations built up from underlying values, then comonads can be seen as reductions back down to values. Monadic code, in a sense, cannot be fully "unpacked"; once a value is wrapped within a monad, it remains quarantined there along with any side-effects (a good thing in purely functional programming). Sometimes though, a problem is more about consuming contextual data, which comonads can model explicitly.

Technically, a comonad is the categorical dual of a monad, which loosely means that it will have the same required components, only with the direction of the type signatures reversed. Starting from the bind-centric monad definition, a comonad consists of:

• A type constructor W that marks the higher-order type W T
• The dual of unit, called counit here, extracts the underlying value from the comonad:

counit(wa) : W T → T

• A reversal of bind (also represented with =>>) that extends a chain of reducing functions:

(wa =>> f) : (W U, W U → T) → W T[lower-alpha 18]

extend and counit must also satisfy duals of the monad laws:

counit ∘ ( (wa =>> f) → wb )  ↔  f(wa) → b
wa =>> counit  ↔  wa
wa ( (=>> f(wx = wa)) → wb (=>> g(wy = wb)) → wc )  ↔  ( wa (=>> f(wx = wa)) → wb ) (=>> g(wy = wb)) → wc

Analogous to monads, comonads can also be derived from functors using a dual of join:

• duplicate takes an already comonadic value and wraps it in another layer of comonadic structure:

duplicate(wa) : W T → W (W T)

While operations like extend are reversed, however, a comonad does not reverse functions it acts on, and consequently, comonads are still functors with map, not cofunctors. The alternate definition with duplicate, counit, and map must also respect its own comonad laws:

((map duplicate) ∘ duplicate) wa  ↔  (duplicate ∘ duplicate) wa  ↔  wwwa
((map counit) ∘ duplicate)    wa  ↔  (counit ∘ duplicate)    wa  ↔  wa
((map map φ) ∘ duplicate)     wa  ↔  (duplicate ∘ (map φ))   wa  ↔  wwb

And as with monads, the two forms can be converted automatically:

(map φ) wa    ↔  wa =>> (φ ∘ counit) wx
duplicate wa  ↔  wa =>> wx

wa =>> f(wx)  ↔  ((map f) ∘ duplicate) wa

A simple example is the Product comonad, which outputs values based on an input value and shared environment data. In fact, the Product comonad is just the dual of the Writer monad and effectively the same as the Reader monad (both discussed below). Product and Reader differ only in which function signatures they accept, and how they complement those functions by wrapping or unwrapping values.

A less trivial example is the Stream comonad, which can be used to represent data streams and attach filters to the incoming signals with extend. In fact, while not as popular as monads, researchers have found comonads particularly useful for stream processing and modeling dataflow programming.^[36][37]

Due to their strict definitions, however, one cannot simply move objects back and forth between monads and comonads. As an even higher abstraction, arrows can subsume both structures, but finding more granular ways to combine monadic and comonadic code is an active area of research.^[38][39]