nLab monad (in computer science)



Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels


Constructivism, Realizability, Computability



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

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

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

  • is in a way that is associative in the evident sense and unital with respect to a given unit function called pure X:XT(X)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 the sense of 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 an account of the use of monads in industry see Benton15, pp. 11-12.

For instance when the monad T()T(-) forms product types T(X)X×QT(X) \coloneqq X \times Q with some fixed type QQ that carries the structure of a monoid, then a Kleisli function f:XY×Qf : X \to Y \times Q may be thought of as a function XYX \to Y that produces a side effect output of type QQ. The Kleisli composition of two functions f:XY×Qf \colon X \to Y \times Q and g:YZ×Qg \colon Y \to Z \times Q then not only evaluates the two programs in sequence but also combines their QQ-output using the monoid operation of QQ; so if fx=(y,q)f x = (y,q) and gy=(z,q)g y = (z,q') then the final result of (gf)(x)(g \circ f)(x) will be (z,qq)(z, q q'). For example, QQ might be the set of strings of characters, and the monoid operation that of concatenation of strings (i.e. QQ is the free monoid on the type of characters). If the software is designed such that values of type QQ 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 𝒞\mathcal{C}, whose

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:𝒞𝒞 T \colon \mathcal{C} \longrightarrow \mathcal{C}

on the syntactic category. This functor

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

  2. the unit natural transformation ϵ:Id 𝒞T\epsilon \colon Id_{\mathcal{C}} \Rightarrow T of the monad TT provides for each type XX a component morphism pure X:XT(X)pure_X : X \to T(X);

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

(gf) (YgT(Z)) Kleisli(XfT(Y)) XfT(Y)T(g)T(T(Z))μ(Z)TZ, \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)T(g) in the middle of the last line makes use of the fact that T()T(-) is indeed a functor and hence may also be applied to morphisms / functions between types. The last morphism μ(Z)\mu(Z) is the one that implements the “TT-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 𝒞\mathcal{C} will be at least monoidal, and the corresponding kind of ‘nice’ interaction corresponds to the monad’s being a strong monad.

The ‘bind’ operation is a means of describing multiplication on such a strong monad MM. It is a term of the form MA(MB) AMBM A \to (M B)^A \to M B, which is equivalent to a map of the form MA×MB AMBM A \times M B^A \to M B. It is the composite

MA×MB AstrengthM(A×MB A)Meval A,MBMMBmBMB M A \times M B^A \stackrel{strength}{\to} M(A \times M B^A) \stackrel{M eval_{A, M B}}{\to} M M B \stackrel{m B}{\to} M B

where m:MMMm \colon M M \to M is the monad multiplication.

When monads are defined in Haskell, the Kleisli composition, ‘bind’, is defined in Haskell. So monads in Haskell are always enriched monads, according to the self-enrichment defined by the function type in Haskell.


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 XX, output of type YY and mutable state SS is a function (morphism) of type X×SY×SX \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[S,S×Y]X \longrightarrow [S, S \times Y ]. Here the operation [S,S×()][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×()][S, S\times (-)] is called the state monad for mutable states of type S.

  • The maybe monad is the operation XX*X \mapsto X \coprod \ast. The idea here is that a function XYX \longrightarrow Y in its Kleisli category is in the original category a function of the form XY*X \longrightarrow Y \coprod \ast so either returns indeed a value in YY 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 SS acts by X[[X,S],S]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 comonad.

    Given a type WW, the reader monad is the operation of forming the function type [W,]=(W())[W,-] = (W\to (-)); the writer monad and the writer comonad are the operations of forming the product type W×()W\times (-), and the composite of writer followed by reader is the state monad [W,W×()][W, W \times (-)].

    When WW carries the structure of a monoid object then writer also inherits the structure of a monad (on top of being a comonad) and converse for reader.

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

This includes

See also:

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

See also

For an approach to composing monads, see

Another approach to modelling side effects in functional programming languages are

Free monads in computer science appear in the concepts of

Other generalizations are

There is also



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

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

The impact of Moggi’s work is assessed and a case for Lawvere theories is made in

  • Martin Hyland, John Power, The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads, Electronic Notes in Theoretical Computer Science (ENTCS) archive Volume 172, April, 2007 Pages 437-458 (pdf)

Effects treated this way are known as algebraic effects.

Expositions of monads in computer science include

  • 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

  • Philip Mulry, Monads in semantics , ENTCS 14 (1998) pp.275-286.

  • 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)

  • Nick Benton, Categorical Monads and Computer Programming, (pdf)

  • Emily Riehl, A categorical view of computational effects, 2017 (pdf)

See also:

The specification of monads in Haskell is at

and an exposition of category theory and monads in terms of Haskell is in

A comparison of monads with applicative functors (also known as idioms) and with arrows (in computer science) is in

  • Exequiel Rivas, Relating Idioms, Arrows and Monads from Monoidal Adjunctions, (arXiv:1807.04084)

Generalization from monads to relative monads is discussed in

Discussion of comonads in this context includes

  • David Overton, Comonads in Haskell, 2014 (web)

In quantum computation

Discussion of aspects of quantum computing in terms of monads in functional programming are in

Last revised on December 14, 2020 at 13:06:50. See the history of this page for a list of all contributions to it.