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In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, a branch of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a monad (also triple, triad, standard construction and fundamental construction) is a
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
in the
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
of
endofunctor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
s. An endofunctor is a
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
mapping a category to itself, and a monad is an endofunctor together with two
natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
s required to fulfill certain
coherence condition In mathematics, and particularly category theory, a coherence condition is a collection of conditions requiring that various compositions of elementary morphisms are equal. Typically the elementary morphisms are part of the data of the category. A ...
s. Monads are used in the theory of pairs of
adjoint functors In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
, and they generalize
closure operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S : Closure operators are dete ...
s on
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
s to arbitrary categories. Monads are also useful in the theory of datatypes and in
functional programming language In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in which function definitions are trees of expressions that m ...
s, allowing languages with non-mutable states to do things such as simulate for-loops; see
Monad (functional programming) In functional programming, a monad is a software design pattern with a structure that combines program fragments (Function (computer programming), functions) and wraps their return values in a Type system, type with additional computation. In add ...
.


Introduction and definition

A monad is a certain type of
endofunctor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
. For example, if F and G are a pair of
adjoint functors In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
, with F left adjoint to G, then the composition G \circ F is a monad. If F and G are inverse functors, the corresponding monad is the
identity functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
. In general, adjunctions are not
equivalences Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry *Equivalence class (music) *''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *'' Equival ...
—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 F \circ G, is discussed under the dual theory of ''comonads''.


Formal definition

Throughout this article C denotes a
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
. A ''monad'' on C consists of an endofunctor T \colon C \to C together with two
natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
s: \eta \colon 1_ \to T (where 1_ denotes the identity functor on C) and \mu \colon T^ \to T (where T^ is the functor T \circ T from C to C). These are required to fulfill the following conditions (sometimes called
coherence condition In mathematics, and particularly category theory, a coherence condition is a collection of conditions requiring that various compositions of elementary morphisms are equal. Typically the elementary morphisms are part of the data of the category. A ...
s): * \mu \circ T\mu = \mu \circ \mu T (as natural transformations T^ \to T); here T\mu and \mu T are formed by " horizontal composition" * \mu \circ T \eta = \mu \circ \eta T = 1_ (as natural transformations T \to T; here 1_ denotes the identity transformation from T to T). We can rewrite these conditions using the following
commutative diagrams 350px, The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to th ...
: See the article on
natural transformations In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural ...
for the explanation of the notations T\mu and \mu T, or see below the commutative diagrams not using these notions: The first axiom is akin to the
associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
in
monoids In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
if we think of \mu as the monoid's binary operation, and the second axiom is akin to the existence of an
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
(which we think of as given by \eta). Indeed, a monad on C can alternatively be defined as a
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
in the category \mathbf_ whose objects are the endofunctors of C and whose morphisms are the natural transformations between them, with the monoidal structure induced by the composition of endofunctors.


The power set monad

The ''power set monad'' is a monad \mathcal on the category \mathbf: For a set A let T(A) be the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of A and for a function f \colon A \to B let T(f) be the function between the power sets induced by taking direct images under f. For every set A, we have a map \eta_ \colon A \to T(A), which assigns to every a\in A the
singleton Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance ...
\. The function :\mu_ \colon T(T(A)) \to T(A) takes a set of sets to its
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
. These data describe a monad.


Remarks

The axioms of a monad are formally similar to the
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
axioms. In fact, monads are special cases of monoids, namely they are precisely the monoids among
endofunctor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
s \operatorname(C), 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 \mathcal \circ \mathcal does not admit any monad structure.


Comonads

The
categorical dual In category theory, a branch of mathematics, duality is a correspondence between the properties of a category ''C'' and the dual properties of the opposite category ''C''op. Given a statement regarding the category ''C'', by interchanging the sou ...
definition is a formal definition of a ''comonad'' (or ''cotriple''); this can be said quickly in the terms that a comonad for a category C is a monad for the
opposite category In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields t ...
C^. It is therefore a functor U from C 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 In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
than monoids; however, comonoids in the category of vector spaces with its usual tensor product are important and widely studied under the name of
coalgebra In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams ...
s.


Terminological history

The notion of monad was invented by
Roger Godement Roger Godement (; 1 October 1921 – 21 July 2016) was a French mathematician, known for his work in functional analysis as well as his expository books. Biography Godement started as a student at the École normale supérieure in 1940, where he ...
in 1958 under the name "standard construction". Monad has been called "dual standard construction", "triple", "monoid" and "triad". The term "monad" is used at latest 1967, by
Jean Bénabou Jean Bénabou (1932 – 11 February 2022) was a Moroccan-born French mathematician, known for his contributions to category theory. He directed the Research Seminar in Category Theory at the Institut Henri Poincaré The Henri Poincaré Insti ...
.


Examples


Monads arising from adjunctions

Any adjunction :F: C \rightleftarrows D : G gives rise to a monad on ''C''. This very widespread construction works as follows: the endofunctor is the composite :T = G \circ F. This endofunctor is quickly seen to be a monad, where the unit map stems from the unit map \operatorname_C \to G \circ F of the adjunction, and the multiplication map is constructed using the counit map of the adjunction: :T^2 = G \circ F \circ G \circ F \xrightarrow G \circ F = T. In fact, any monad can be found as an explicit adjunction of functors using the
Eilenberg–Moore category In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is a monoid in the category of endofunctors. An endofunctor is a functor mapping a category to itself, and a monad is ...
C^T (the category of T-algebras).


Double dualization

The ''double dualization monad'', for a fixed
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
''k'' arises from the adjunction :(-)^* : \mathbf_k \rightleftarrows \mathbf_k^ : (-)^* where both functors are given by sending a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
''V'' to its
dual vector space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
V^* := \operatorname(V, k). The associated monad sends a vector space ''V'' to its
double dual In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
V^. This monad is discussed, in much greater generality, by .


Closure operators on partially ordered sets

For categories arising from
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
s (P, \le) (with a single morphism from x to y
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
x \le y), then the formalism becomes much simpler: adjoint pairs are
Galois connection In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the funda ...
s and monads are
closure operators Closure may refer to: Conceptual Psychology * Closure (psychology), the state of experiencing an emotional conclusion to a difficult life event Computer science * Closure (computer programming), an abstraction binding a function to its scope * ...
.


Free-forgetful adjunctions

For example, let G be the
forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signa ...
from the category Grp of
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
to the category Set of sets, and let F be the
free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
functor from the category of sets to the category of groups. Then F is left adjoint of G. In this case, the associated monad T = G \circ F takes a set X and returns the underlying set of the free group \mathrm(X). The unit map of this monad is given by the maps :X \to T(X) including any set X into the set \mathrm(X) in the natural way, as strings of length 1. Further, the multiplication of this monad is the map :T(T(X)) \to T(X) made out of a natural
concatenation In formal language, formal language theory and computer programming, string concatenation is the operation of joining character string (computer science), character strings wikt:end-to-end, end-to-end. For example, the concatenation of "sno ...
or 'flattening' of 'strings of strings'. This amounts to two
natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
s. 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 Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, ...
. 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 T is the endofunctor on the category of vector spaces which maps a vector space V to its
tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
T(V), and which maps linear maps to their tensor product. We then have a natural transformation corresponding to the embedding of V into its
tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
, and a natural transformation corresponding to the map from T(T(V)) to T(V) obtained by simply expanding all tensor products.


Codensity monads

Under mild conditions, functors not admitting a left adjoint also give rise to a monad, the so-called codensity monad. For example, the inclusion :\mathbf \subset \mathbf does not admit a left adjoint. Its codensity monad is the monad on sets sending any set ''X'' to the set of
ultrafilter In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on ...
s on ''X''. This and similar examples are discussed in .


Algebras for a monad

Given a monad (T,\eta,\mu) on a category C, it is natural to consider ''T-algebras'', i.e., objects of C acted upon by T in a way which is compatible with the unit and multiplication of the monad. More formally, a T-algebra (x,h) is an object x of C together with an arrow h\colon Tx\to x of C called the ''structure map'' of the algebra such that the diagrams commute. A morphism f\colon (x,h)\to(x',h') of T-algebras is an arrow f\colon x\to x' of C such that the diagram commutes. T-algebras form a category called the ''Eilenberg–Moore category'' and denoted by C^T.


Examples


Algebras over the free group monad

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


Algebras over the distribution monad

Another example is the ''distribution monad'' \mathcal on the category of sets. It is defined by sending a set X to the set of functions f : X \to ,1/math> with finite support and such that their sum is equal to 1. In set-builder notation, this is the set\mathcal(X) = \left\By inspection of the definitions, it can be shown that algebras over the distribution monad are equivalent to
convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...
s, i.e., sets equipped with operations x +_r y for r \in ,1/math> subject to axioms resembling the behavior of convex linear combinations rx + (1-r)y in Euclidean space.


Algebras over the symmetric monad

Another useful example of a monad is the symmetric algebra functor on the category of R-modules for a commutative ring R.\text^\bullet(-): \text(R) \to \text(R)sending an R-module M to the direct sum of
symmetric tensor In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments: :T(v_1,v_2,\ldots,v_r) = T(v_,v_,\ldots,v_) for every permutation ''σ'' of the symbols Alternatively, a symmetric tensor of orde ...
powers\text^\bullet(M) = \bigoplus_^\infty \text^k(M)where \text^0(M) = R. For example, \text^\bullet(R^) \cong R _1,\ldots, x_n/math> where the R-algebra on the right is considered as a module. Then, an algebra over this monad are commutative R-algebras. There are also algebras over the monads for the alternating tensors \text^\bullet(-) and total tensor functors T^\bullet(-) giving anti-symmetric R-algebras, and free R-algebras, so\begin \text^\bullet(R^) &= R(x_1,\ldots, x_n)\\ \text^\bullet(R^) &= R\langle x_1,\ldots, x_n \rangle \endwhere the first ring is the free anti-symmetric algebra over R in n-generators and the second ring is the free algebra over R in n-generators.


Commutative algebras in E-infinity ring spectra

There is an analogous construction for commutative \mathbb-algebraspg 113 which gives commutative A-algebras for a commutative \mathbb-algebra A. If \mathcal_A is the category of A-modules, then the functor \mathbb: \mathcal_A \to \mathcal_A is the monad given by\mathbb(M) = \bigvee_ M^j/\Sigma_jwhereM^j = M\wedge_A \cdots \wedge_A M j-times. Then there is an associated category \mathcal_A of commutative A-algebras from the category of algebras over this monad.


Monads and adjunctions

As was mentioned above, any adjunction gives rise to a monad. Conversely, every monad arises from some adjunction, namely the free–forgetful adjunction :T(-) : C \rightleftarrows C^T : \text 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 \mathbf(C,T) be the category whose objects are the adjunctions (F,G,e,\varepsilon) such that (GF, e, G\varepsilon F)=(T,\eta,\mu) and whose arrows are the morphisms of adjunctions that are the identity on C. Then the above free–forgetful adjunction involving the Eilenberg–Moore category C^T is a terminal object in \mathbf(C,T). An initial object is the
Kleisli category In category theory, a Kleisli category is a category naturally associated to any monad ''T''. It is equivalent to the category of free ''T''-algebras. The Kleisli category is one of two extremal solutions to the question ''Does every monad arise fr ...
, which is by definition the full subcategory of C^T consisting only of free ''T''-algebras, i.e., ''T''-algebras of the form T(x) for some object ''x'' of ''C''.


Monadic adjunctions

Given any adjunction (F : C \to D,G : D \to C,\eta,\varepsilon) with associated monad ''T'', the functor ''G'' can be factored as :D \stackrel \to C^T \stackrel \to C, 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 \tilde G yields an
equivalence of categories In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences fro ...
between ''D'' and the Eilenberg–Moore category C^T. By extension, a functor G\colon D\to C is said to be monadic if it has a left adjoint F 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

''
Beck's monadicity theorem In category theory, a branch of mathematics, Beck's monadicity theorem gives a criterion that characterises monadic functors, introduced by in about 1964. It is often stated in dual form for comonads. It is sometimes called the Beck tripleabi ...
'' 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 Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization i ...
(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
coequalizer In category theory, a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. It is the categorical construction dual to the equalizer. Definition A coequalizer is a co ...
s. For example, the forgetful functor from the category of
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
s 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 In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorp ...
. Thus, this forgetful functor is not monadic. The dual version of Beck's theorem, characterizing comonadic adjunctions, is relevant in different fields such as
topos theory In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion ...
and topics in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
related to
descent Descent may refer to: As a noun Genealogy and inheritance * Common descent, concept in evolutionary biology * Kinship, one of the major concepts of cultural anthropology **Pedigree chart or family tree ** Ancestry ** Lineal descendant **Heritag ...
. A first example of a comonadic adjunction is the adjunction :- \otimes_A B : \mathbf_A \rightleftarrows \mathbf_B : \operatorname for a ring homomorphism A \to B 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 Faithfully flat descent is a technique from algebraic geometry, allowing one to draw conclusions about objects on the target of a faithfully flat morphism. Such morphisms, that are flat and surjective, are common, one example coming from an open c ...
is widely applied in algebraic geometry.


Uses

Monads are used in
functional programming In computer science, functional programming is a programming paradigm where programs are constructed by Function application, applying and Function composition (computer science), composing Function (computer science), functions. It is a declar ...
to express types of sequential computation (sometimes with side-effects). See
monads in functional programming In functional programming, a monad is a software design pattern with a structure that combines program fragments ( functions) and wraps their return values in a type with additional computation. In addition to defining a wrapping monadic type, m ...
, 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 Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...
via
closure operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S : Closure operators are dete ...
s,
interior algebra In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and or ...
s, and their relation to
models A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
of S4 and
intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
s.


Generalization

It is possible to define monads in a
2-category In category theory, a strict 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over Cat (the category of catego ...
C. Monads described above are monads for C = \mathbf.


See also

* Distributive law between monads *
Lawvere theory In category theory, a Lawvere theory (named after United States, American mathematician William Lawvere) is a category (mathematics), category that can be considered a categorical counterpart of the notion of an equational theory. Definition Let \ ...
*
Monad (functional programming) In functional programming, a monad is a software design pattern with a structure that combines program fragments (Function (computer programming), functions) and wraps their return values in a Type system, type with additional computation. In add ...
* Polyad * Strong monad


References


Further reading

* * * * * * * * {{Citation, first=Daniele, last=Turi, url=http://www.dcs.ed.ac.uk/home/dt/CT/categories.pdf, title=Category Theory Lecture Notes, year=1996–2001


External links


Monads
five short lectures (with one appendix). *John Baez'

covers monads in 2-categories. Adjoint functors Category theory