# Monad (category theory)

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In category theory, a branch of mathematics, a **monad** (also **triple**, **triad**, **standard construction** and **fundamental construction**)^{[1]} is an (endo-)functor, together with two natural transformations. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories.

## Introduction

If and are a pair of adjoint functors, with left adjoint to , then the composition is a monad. Therefore, a monad is an endofunctor. If and are inverse functors, the corresponding monad is the identity functor. In general, adjunctions are not equivalences — they relate categories of different natures. The monad theory matters as part of the effort to capture what it is that adjunctions 'preserve'. The other half of the theory, of what can be learned likewise from consideration of , is discussed under the dual theory of **comonads**.

The monad axioms can be seen at work in a simple example: let be the forgetful functor from the category **Grp** of groups to the category **Set** of sets. Then as we can take the free group functor.

This means that the monad

takes a set and returns the underlying set of the free group . In this situation, we are given two natural morphisms:

by including any set in in the natural way, as strings of length 1. Further,

can be made out of a natural concatenation or 'flattening' of 'strings of strings'. This amounts to two natural transformations

and

They will satisfy some axioms about identity and associativity that result from the adjunction properties.

Those axioms are formally similar to the monoid axioms. They are taken as the definition of a general monad (not assumed *a priori* to be connected to an adjunction) on a category.

If we specialize to categories arising from partially ordered sets (with a single morphism from to iff ), then the formalism becomes much simpler: adjoint pairs are Galois connections and monads are closure operators.

Every monad arises from some adjunction, in fact typically from many adjunctions. Two constructions introduced below, the **Kleisli category** and the category of **Eilenberg-Moore algebras**, are extremal solutions of the problem of constructing an adjunction that gives rise to a given monad.

The example about free groups given above can be generalized to any type of algebra in the sense of a variety of algebras in universal algebra. Thus, every such type of algebra gives rise to a monad on the category of sets. Importantly, the algebra type can be recovered from the monad (as the category of Eilenberg-Moore algebras), so monads can also be seen as generalizing universal algebras. Even more generally, any adjunction is said to be **monadic** (or **tripleable**) if it shares this property of being (equivalent to) the Eilenberg-Moore category of its associated monad. Consequently Beck's monadicity theorem, which gives a criterion for monadicity, can be used to show that an arbitrary adjunction can be treated as a category of algebras in this way.

The notion of monad was invented by Roger Godement in 1958 under the name "standard construction." In the 1960s and 1970s, many people used the name "triple." The now standard term "monad" is due to Mac Lane.

## Formal definition

If is a category, a **monad** on consists of a functor together with two natural transformations: (where denotes the identity functor on ) and (where is the functor from to ). These are required to fulfill the following conditions (sometimes called coherence conditions):

- (as natural transformations );
- (as natural transformations ; here denotes the identity transformation from to ).

We can rewrite these conditions using following commutative diagrams:

See the article on natural transformations for the explanation of the notations and , or see below the commutative diagrams not using these notions:

The first axiom is akin to the associativity in monoids, the second axiom to the existence of an identity element. Indeed, a monad on can alternatively be defined as a monoid in the category whose objects are the endofunctors of and whose morphisms are the natural transformations between them, with the monoidal structure induced by the composition of endofunctors.

## Comonads and their importance

The *categorical dual* definition is a formal definition of a **comonad** (or **cotriple**); this can be said quickly in the terms that a comonad for a category is a monad for the opposite category . It is therefore a functor from to itself, with a set of axioms for *counit* and *comultiplication* that come from reversing the arrows everywhere in the definition just given.

Since a *comonoid* is not a basic structure in abstract algebra, this is less familiar at an immediate level.

The importance of the definition comes in a class of theorems from the categorical (and algebraic geometry) theory of descent. What was realised in the period 1960 to 1970 is that *recognising* the categories of *coalgebras for a comonad* was an important tool of category theory (particularly topos theory). The results involved are based on Beck's theorem. Roughly what goes on is this: while it is simple set theory that a surjective mapping of sets is as good as the equivalence relation "*x* is in the same fiber as *y*" on the domain of the mapping, for geometric morphisms what you should do is pass to such a coalgebra subcategory.

## Examples

The rich set of examples is given by adjunctions (see Monads and adjunctions), and the free group example mentioned above belongs to that set.

Here is another example, on the category : For a set let be the power set of and for a function let be the function between the power sets induced by taking direct images under . For every set , we have a map , which assigns to every the singleton . The function

takes a set of sets to its union. These data describe a monad.

Closure operators are monads on preorder categories.

## Algebras for a monad

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Suppose that is a given monad on a category .

A **-algebra** is an object of together with an arrow of called the **structure map** of the algebra such that the diagrams

and |

commute.

A morphism of -algebras is an arrow of such that the diagram

commutes.

The category of -algebras and their morphisms is called the **Eilenberg-Moore category** or **category of (Eilenberg-Moore) algebras** of the monad . The forgetful functor → has a left adjoint → taking to the free algebra .

Given the monad , there exists another "canonical" category called the **Kleisli category** of the monad . This category is equivalent to the *category of free algebras* for the monad , i. e. the full subcategory of whose objects are of the form , for an object of .

## Monads and adjunctions

An adjunction between two categories and (where is left adjoint to and and are respectively the unit and the counit) always defines a monad .

Conversely, it is interesting to consider the adjunctions which define a given monad this way. Let be the category whose objects are the adjunctions such that and whose arrows are the morphisms of adjunctions which are the identity on . Then this category has

- an initial object , where is the Kleisli category,
- a terminal object , where is the Eilenberg-Moore category.

An adjunction between two categories and is a **monadic adjunction** when the category is equivalent to the Eilenberg-Moore category for the monad . By extension, a functor is said to be **monadic** if it has a left adjoint forming a monadic adjunction. Beck's monadicity theorem gives a characterization of monadic functors.

## Uses

Monads are used in functional programming to express types of sequential computation (sometimes with side-effects). See monads in functional programming, and the more mathematically oriented Wikibook module b:Haskell/Category theory.

In categorical logic, an analogy has been drawn between the monad-comonad theory, and modal logic via closure operators, interior algebras, and their relation to models of S4 and Intuitionistic logics.

## Generalization

It is possible to define monads in a 2-category . Monads described above are monads for .

## See also

- Distributive law between monads
- Strong monad
- Monad (disambiguation) for other meanings of the term.
- Monad (functional programming)
- Polyad

## References

- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}

## Further reading

- Daniele Turi:
*Category Theory Lecture Notes*(1996-2001), based on MacLane's book "Categories for the Working Mathematician" - Michael Barr and Charles Wells:
*Category Theory for Computing Science*(1999). - Roger Godement: Topologie Algébrique et Théorie des Faisceaux. Actualités Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13 Hermann, Paris 1958 viii+283 pp.
- {{#invoke:citation/CS1|citation

|CitationClass=book }}

## External links

- Monads, five short lectures (with one appendix).
- John Baez's This Week's Finds in Mathematical Physics (Week 89) covers monads in 2-categories.