Monad (category theory)

In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction)[1] is a monoid in the category of endofunctors of some fixed category. An endofunctor is a functor mapping a category to itself, and a monad is an endofunctor together with two natural transformations required to fulfill certain coherence conditions. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories. Monads are also useful in the theory of datatypes, the denotational semantics of imperative programming languages, and in functional programming languages, allowing languages without mutable state to do things such as simulate for-loops; see Monad (functional programming).

Introduction and definition

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A monad is a certain type of endofunctor. For example, if   and   are a pair of adjoint functors, with   left adjoint to  , then the composition   is a monad. 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.

Formal definition

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Throughout this article   denotes a category. A monad on   consists of an endofunctor   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  ); here   and   are formed by "horizontal composition."
  •   (as natural transformations  ; here   denotes the identity transformation from   to  ).

We can rewrite these conditions using the 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 if we think of   as the monoid's binary operation, and the second axiom is akin to the existence of an identity element (which we think of as given by  ). 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.

The power set monad

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The power set monad is a monad   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.

Remarks

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The axioms of a monad are formally similar to the monoid axioms. In fact, monads are special cases of monoids, namely they are precisely the monoids among endofunctors  , which is equipped with the multiplication given by composition of endofunctors.

Composition of monads is not, in general, a monad. For example, the double power set functor   does not admit any monad structure.[2]

Comonads

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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.

Monads are to monoids as comonads are to comonoids. Every set is a comonoid in a unique way, so comonoids are less familiar in abstract algebra than monoids; however, comonoids in the category of vector spaces with its usual tensor product are important and widely studied under the name of coalgebras.

Terminological history

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The notion of monad was invented by Roger Godement in 1958 under the name "standard construction". Monad has been called "dual standard construction", "triple", "monoid" and "triad".[3] The term "monad" is used at latest 1967, by Jean Bénabou.[4][5]

Examples

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Identity

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The identity functor on a category   is a monad. Its multiplication and unit are the identity function on the objects of  .

Monads arising from adjunctions

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Any adjunction

 

gives rise to a monad on C. This very widespread construction works as follows: the endofunctor is the composite

 

This endofunctor is quickly seen to be a monad, where the unit map stems from the unit map   of the adjunction, and the multiplication map is constructed using the counit map of the adjunction:

 

In fact, any monad can be found as an explicit adjunction of functors using the Eilenberg–Moore category   (the category of  -algebras).[6]

Double dualization

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The double dualization monad, for a fixed field k arises from the adjunction

 

where both functors are given by sending a vector space V to its dual vector space  . The associated monad sends a vector space V to its double dual  . This monad is discussed, in much greater generality, by Kock (1970).

Closure operators on partially ordered sets

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

Free-forgetful adjunctions

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For example, let   be the forgetful functor from the category Grp of groups to the category Set of sets, and let   be the free group functor from the category of sets to the category of groups. Then   is left adjoint of  . In this case, the associated monad   takes a set   and returns the underlying set of the free group  . The unit map of this monad is given by the maps

 

including any set   into the set   in the natural way, as strings of length 1. Further, the multiplication of this monad is the map

 

made out of a natural concatenation or 'flattening' of 'strings of strings'. This amounts to two natural transformations. The preceding example about free groups 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 varieties of universal algebras.

Another monad arising from an adjunction is when   is the endofunctor on the category of vector spaces which maps a vector space   to its tensor algebra  , and which maps linear maps to their tensor product. We then have a natural transformation corresponding to the embedding of   into its tensor algebra, and a natural transformation corresponding to the map from   to   obtained by simply expanding all tensor products.

Codensity monads

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Under mild conditions, functors not admitting a left adjoint also give rise to a monad, the so-called codensity monad. For example, the inclusion

 

does not admit a left adjoint. Its codensity monad is the monad on sets sending any set X to the set of ultrafilters on X. This and similar examples are discussed in Leinster (2013).

Monads used in denotational semantics

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The following monads over the category of sets are used in denotational semantics of imperative programming languages, and analogous constructions are used in functional programming.

The maybe monad

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The endofunctor of the maybe or partiality monad adds a disjoint point:[7]

 
 

The unit is given by the inclusion of a set   into  :

 
 

The multiplication maps elements of   to themselves, and the two disjoint points in   to the one in  .

In both functional programming and denotational semantics, the maybe monad models partial computations, that is, computations that may fail.

The state monad

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Given a set  , the endofunctor of the state monad maps each set   to the set of functions  . The component of the unit at   maps each element   to the function

 
 

The multiplication maps the function   to the function

 
 

In functional programming and denotational semantics, the state monad models stateful computations.

The environment monad

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Given a set  , the endofunctor of the reader or environment monad maps each set   to the set of functions  . Thus, the endofunctor of this monad is exactly the hom functor  . The component of the unit at   maps each element   to the constant function  .

In functional programming and denotational semantics, the environment monad models computations with access to some read-only data.

The list and set monads

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The list or nondeterminism monad maps a set X to the set of finite sequences (i.e., lists) with elements from X. The unit maps an element x in X to the singleton list [x]. The multiplication concatenates a list of lists into a single list.

In functional programming, the list monad is used to model nondeterministic computations. The covariant powerset monad is also known as the set monad, and is also used to model nondeterministic computation.

Algebras for a monad

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Given a monad   on a category  , it is natural to consider  -algebras, i.e., objects of   acted upon by   in a way which is compatible with the unit and multiplication of the monad. More formally, 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.  -algebras form a category called the Eilenberg–Moore category and denoted by  .

Examples

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Algebras over the free group monad

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For example, for the free group monad discussed above, a  -algebra is a set   together with a map from the free group generated by   towards   subject to associativity and unitality conditions. Such a structure is equivalent to saying that   is a group itself.

Algebras over the distribution monad

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Another example is the distribution monad   on the category of sets. It is defined by sending a set   to the set of functions   with finite support and such that their sum is equal to  . In set-builder notation, this is the set By inspection of the definitions, it can be shown that algebras over the distribution monad are equivalent to convex sets, i.e., sets equipped with operations   for   subject to axioms resembling the behavior of convex linear combinations   in Euclidean space.[8]

Algebras over the symmetric monad

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Another useful example of a monad is the symmetric algebra functor on the category of  -modules for a commutative ring  . sending an  -module   to the direct sum of symmetric tensor powers where  . For example,   where the  -algebra on the right is considered as a module. Then, an algebra over this monad are commutative  -algebras. There are also algebras over the monads for the alternating tensors   and total tensor functors   giving anti-symmetric  -algebras, and free  -algebras, so where the first ring is the free anti-symmetric algebra over   in  -generators and the second ring is the free algebra over   in  -generators.

Commutative algebras in E-infinity ring spectra

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There is an analogous construction for commutative  -algebras[9]pg 113 which gives commutative  -algebras for a commutative  -algebra  . If   is the category of  -modules, then the functor   is the monad given by where   -times. Then there is an associated category   of commutative  -algebras from the category of algebras over this monad.

Monads and adjunctions

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As was mentioned above, any adjunction gives rise to a monad. Conversely, every monad arises from some adjunction, namely the free–forgetful adjunction

 

whose left adjoint sends an object X to the free T-algebra T(X). However, there are usually several distinct adjunctions giving rise to a monad: let   be the category whose objects are the adjunctions   such that   and whose arrows are the morphisms of adjunctions that are the identity on  . Then the above free–forgetful adjunction involving the Eilenberg–Moore category   is a terminal object in  . An initial object is the Kleisli category, which is by definition the full subcategory of   consisting only of free T-algebras, i.e., T-algebras of the form   for some object x of C.

Monadic adjunctions

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Given any adjunction   with associated monad T, the functor G can be factored as

 

i.e., G(Y) can be naturally endowed with a T-algebra structure for any Y in D. The adjunction is called a monadic adjunction if the first functor   yields an equivalence of categories between D and the Eilenberg–Moore category  .[10] By extension, a functor   is said to be monadic if it has a left adjoint   forming a monadic adjunction. For example, the free–forgetful adjunction between groups and sets is monadic, since algebras over the associated monad are groups, as was mentioned above. In general, knowing that an adjunction is monadic allows one to reconstruct objects in D out of objects in C and the T-action.

Beck's monadicity theorem

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Beck's monadicity theorem gives a necessary and sufficient condition for an adjunction to be monadic. A simplified version of this theorem states that G is monadic if it is conservative (or G reflects isomorphisms, i.e., a morphism in D is an isomorphism if and only if its image under G is an isomorphism in C) and C has and G preserves coequalizers.

For example, the forgetful functor from the category of compact Hausdorff spaces to sets is monadic. However the forgetful functor from all topological spaces to sets is not conservative since there are continuous bijective maps (between non-compact or non-Hausdorff spaces) that fail to be homeomorphisms. Thus, this forgetful functor is not monadic.[11] The dual version of Beck's theorem, characterizing comonadic adjunctions, is relevant in different fields such as topos theory and topics in algebraic geometry related to descent. A first example of a comonadic adjunction is the adjunction

 

for a ring homomorphism   between commutative rings. This adjunction is comonadic, by Beck's theorem, if and only if B is faithfully flat as an A-module. It thus allows to descend B-modules, equipped with a descent datum (i.e., an action of the comonad given by the adjunction) to A-modules. The resulting theory of faithfully flat descent is widely applied in algebraic geometry.

Uses

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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.

Monads are used in the denotational semantics of impure functional and imperative programming languages.[12][13]

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

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

See also

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References

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  1. ^ Barr, Michael; Wells, Charles (1985), "Toposes, Triples and Theories" (PDF), Grundlehren der mathematischen Wissenschaften, vol. 278, Springer-Verlag, pp. 82 and 120, ISBN 0-387-96115-1.
  2. ^ Klin; Salamanca (2018), "Iterated Covariant Powerset is not a Monad", Electronic Notes in Theoretical Computer Science, 341: 261–276, doi:10.1016/j.entcs.2018.11.013
  3. ^ MacLane 1978, p. 138.
  4. ^ Bénabou, Jean (1967). "Introduction to bicategories". In Bénabou, J.; Davis, R.; Dold, A.; Isbell, J.; MacLane, S.; Oberst, U.; Roos, J. -E. (eds.). Reports of the Midwest Category Seminar. Lecture Notes in Mathematics. Vol. 47. Berlin, Heidelberg: Springer. pp. 1–77. doi:10.1007/BFb0074299. ISBN 978-3-540-35545-8.
  5. ^ "RE: Monads". Gmane. 2009-04-04. Archived from the original on 2015-03-26.
  6. ^ Riehl, Emily. "Category Theory in Context" (PDF). p. 162. Archived (PDF) from the original on 5 Apr 2021.
  7. ^ Riehl 2017, p. 155.
  8. ^ Świrszcz, T. (1974), "Monadic functors and convexity", Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys., 22: 39–42, MR 0390019, Jacobs, Bart (2010), "Convexity, Duality and Effects", Theoretical Computer Science, IFIP Advances in Information and Communication Technology, vol. 323, pp. 1–19, doi:10.1007/978-3-642-15240-5_1, ISBN 978-3-642-15239-9
  9. ^ Basterra, M. (1999-12-15). "André–Quillen cohomology of commutative S-algebras". Journal of Pure and Applied Algebra. 144 (2): 111–143. doi:10.1016/S0022-4049(98)00051-6. ISSN 0022-4049.
  10. ^ MacLane (1978) uses a stronger definition, where the two categories are isomorphic rather than equivalent.
  11. ^ MacLane (1978, §§VI.3, VI.9)
  12. ^ Wadler, Philip (1993). "Monads for functional programming". In Broy, Manfred (ed.). Program Design Calculi. NATO ASI Series. Vol. 118. Berlin, Heidelberg: Springer. pp. 233–264. doi:10.1007/978-3-662-02880-3_8. ISBN 978-3-662-02880-3. "The concept of a monad, which arises from category theory, has been applied by Moggi to structure the denotational semantics of programming languages."
  13. ^ Mulry, Philip S. (1998-01-01). "Monads in Semantics". Electronic Notes in Theoretical Computer Science. US-Brazil Joint Workshops on the Formal Foundations of Software Systems. 14: 275–286. doi:10.1016/S1571-0661(05)80241-5. ISSN 1571-0661.

Further reading

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