# Contents

## Idea

In computer science, a monad describes a “notion of computation”. Formally, it is a map that

• sends every type $X$ of some given programming language to a new type $T(X)$ (called the “type of $T$-computations with values in $X$”);

• equipped with a rule for composing two functions of the form $f : X \to T(Y)$ (called Kleisli functions) and $g : Y \to T(Z)$ to a function $g \circ f : X \to T (Z)$ (their Kleisli composition);

• in a way that is associative in the evident sense and unital with respect to a given unit function called $pure_X : X \to T(X)$, to be thought of as taking a value to the pure computation that simply returns that value.

This is essentially the same structure as a monad in category theory, but presented differently; see below for the precise relationship.

### For imperative programs in functional programming

Monads provide one way to “embed imperative programming? in functional programming”, and are used that way notably in the programming language Haskell. (But monads, as well as comonads and related structures, exist much more generally in programming languages; an exposition is in (Harper)).

For instance when the monad $T(-)$ forms product types $T(X) \coloneqq X \times Q$ with some fixed type $Q$ that carries the structure of a monoid, then a Kleisli function $f : X \to Y \times Q$ may be thought of as a function $X \to Y$ that produces a “side effect” output of type $Q$. The Kleisli composition of two functions $f \colon X \to Y \times Q$ and $g \colon Y \to Z \times Q$ then not only evaluates the two programs in sequence but also combines their $Q$-output using the monoid operation of $Q$; so if $f x = (y,q)$ and $g y = (z,q')$ then the final result of $(g \circ f)(x)$ will be $(z, q q')$. For example, $Q$ might be the set of strings of characters, and the monoid operation that of concatenation of strings (i.e. $Q$ is the free monoid on the type of characters). If the software is designed such that values of type $Q$ computed in this way appear on the user’s screen or are written to memory, then this is a way to encode input/output in functional programming (see the IO monad? below).

But monads have plenty of further uses. They are as ubiquituous (sometimes in disguise) in computer science as monads in the sense of category theory are (sometimes in disguise) in category theory. This is no coincidence, see Relation to monads in category theory below.

### Relation to monads in category theory

In computer science, a programming language may be formalised or studied by means of a category, called the syntactic category $\mathcal{C}$, whose

• objects $X \in \mathcal{C}$ are the types of the language,

• morphisms $X \to Y$ are the terms or programs (or an equivalence class of such) that takes a value of type $X$ as input and returns a value of type $Y$.

This point of view (see computational trinitarianism) is particularly useful when studying purely functional programming languages.

Under this relation between type theory and category theory monads on the type system in the sense of computer science are monads in the sense of category theory, being certain endofunctors

$T : \mathcal{C} \to \mathcal{C}$

on the syntactic category. This functor

1. sends each type, hence object $X \in \mathcal{C}$ to another object $T(X)$;

2. the unit natural transformation $\epsilon : Id_{\mathcal{C}} \Rightarrow T$ of the monad $T$ provides for each type $X$ a component morphism $pure_X : X \to T(X)$;

3. the multiplication natural transformation $\mu : T \circ T \Rightarrow T$ of the monad provides for each object $X$ a morphism $\mu_X : T(T(X)) \Rightarrow T(X)$ which induces the Kleisli composition by the formula

\begin{aligned} (g \circ f) &\coloneqq (Y \stackrel{g}{\to} T(Z)) \circ_{Kleisli} (X \stackrel{f}{\to} T(Y)) \\ & \coloneqq X \stackrel{f}{\to} T(Y) \stackrel{T(g)}{\to} T(T(Z)) \stackrel{\mu(Z)}{\to} T Z \end{aligned} \,,

Here the morphism $T(g)$ in the middle of the last line makes use of the fact that $T(-)$ is indeed a functor and hence may also be applied to morphisms / functions between types. The last morphism $\mu(Z)$ is the one that implements the “$T$-computation”.

The monads arising this way in computer science are usually required also to interact nicely with the structure of the programming language, as encoded in the structure of its syntactic category; in most cases, terms of the language will be allowed to take more than one input, so the category $\mathcal{C}$ will be at least monoidal, and the corresponding kind of ‘nice’ interaction corresponds to the monad’s being a strong monad.

## Examples

Various monads are definable in terms of the the standard type-forming operations (product type, function type, etc.). These include the following.

• A functional program with input of type $X$, output of type $Y$ and mutable state $S$ is a function (morphism) of type $X \times S \longrightarrow Y \times S$. Under the (Cartesian product $\dashv$ internal hom)-adjunction this is equivalently given by its adjunct, which is a function of type $X \longrightarrow [S, S \times Y ]$. Here the operation $[S, S\times (-)]$ is the monad induced by the above adjunction and this latter function is naturally regarded as a morphism in the Kleisli category of this monad. This monad $[S, S\times (-)]$ is called the state monad for mutable states of type S.

• The maybe monad is the operation $X \mapsto X \coprod \ast$. The idea here is that a function $X \longrightarrow Y$ in its Kleisli category is in the original category a function of the form $X \longrightarrow Y \coprod \ast$ so either returns indeed a value in $Y$ or else returns the unique element of the unit type/terminal object $\ast$. This is then naturally interpreted as “no value returned”, hence as indicating a “failure in computation”.

• The continuation monad for a given type $S$ acts by $X \mapsto [[X,S],S]$.

• A number of further monads are similarly definable in terms of standard type-forming operations, such as the reader monad? and the writer monad?.

Other monads may be supplied “axiomatically” by the programming language, This includes

Examples of (co)monads in (homotopy) type theory involve in particular modal operators as they appear in

Another approach to modelling side effects in functional programming languages are

Free monads in computer science appear in the concepts of

Another generalization is

## References

### General

The original reference for monads as ‘notions of computation’ is

• Eugenio Moggi, Notions of computation and monads Information And Computation, 93(1), 1991. (pdf)

Expositions are in

• Philip Wadler, Comprehending Monads, in Conference on Lisp and functional programming, ACM Press, 1990 (pdf)

• Philip Wadler, Monads for functional programming in Lecture notes for the Marktoberdorf Summer School on Program Design Calculi, Springer Verlag 1992

• John Hughes, section 2 of Generalising Monads to Arrows, Science of Computer Programming (Elsevier) 37 (1-3): 67–111. (2000) (pdf)

• Robert Harper, Of course ML Has Monads! (2011) (web)