Beck's monadicity theorem
Encyclopedia
In category theory
, a branch of mathematics
, Beck's monadicity theorem asserts that a functor
is monadic if and only if
The second and third condition together can be replaced by a modified condition: every fork in C which is by U sent to a split coequalizer sequence in D
is itself a coequalizer sequence in C. In different words, U creates (preserves and reflects) U-split coequalizer sequences.
This is a basic result of Jonathan Mock Beck from around 1967, often stated in dual form for comonads. It is also sometimes called the Beck tripleability theorem because of the older term triple for a monad.
This theorem is particularly important in its relation with the descent theory
, which plays role in sheaf and stack theory, as well as in the Grothendieck's approach to algebraic geometry
. Most cases of faithfully flat descent of algebraic structure
s (e.g. those in FGA and in SGA1
) are special cases of Beck's theorem. The theorem gives an exact categorical description of the process of 'descent', at this level. In 1970 the Grothendieck approach via fibered categories and descent data was shown (by Bénabou and Roubaud) to be equivalent (under some conditions) to the comonad approach. In a later work, Pierre Deligne
applied Beck's theorem to Tannakian category
theory, greatly simplifying the basic developments.
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Beck's monadicity theorem asserts that a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
is monadic if and only if
- U has a left adjointAdjoint functorsIn mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency...
; - U reflects isomorphismIsomorphismIn abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
s; and - C has coequalizerCoequalizerIn category theory, a coequalizer is a generalization of a quotient by an equivalence relation to objects in an arbitrary category...
s of U-split parallel pairs (those parallel pairs of morphisms in C, which U sends to pairs having a split coequalizer in D), and U preserves those coequalizers.
The second and third condition together can be replaced by a modified condition: every fork in C which is by U sent to a split coequalizer sequence in D
is itself a coequalizer sequence in C. In different words, U creates (preserves and reflects) U-split coequalizer sequences.
This is a basic result of Jonathan Mock Beck from around 1967, often stated in dual form for comonads. It is also sometimes called the Beck tripleability theorem because of the older term triple for a monad.
This theorem is particularly important in its relation with the descent theory
Descent (category theory)
In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of 'gluing' in topology. Since the topologists' glue is actually the use of equivalence relations on topological spaces, the theory starts with some ideas on identification.A sophisticated...
, which plays role in sheaf and stack theory, as well as in the Grothendieck's approach to algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
. Most cases of faithfully flat descent of algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
s (e.g. those in FGA and in SGA1
Grothendieck's Séminaire de géométrie algébrique
In mathematics, the Séminaire de Géométrie Algébrique du Bois Marie was an influential seminar run by Alexander Grothendieck. It was a unique phenomenon of research and publication outside of the main mathematical journals that ran from 1960 to 1969 at the IHÉS near Paris...
) are special cases of Beck's theorem. The theorem gives an exact categorical description of the process of 'descent', at this level. In 1970 the Grothendieck approach via fibered categories and descent data was shown (by Bénabou and Roubaud) to be equivalent (under some conditions) to the comonad approach. In a later work, Pierre Deligne
Pierre Deligne
- See also :* Deligne conjecture* Deligne–Mumford moduli space of curves* Deligne–Mumford stacks* Deligne cohomology* Fourier–Deligne transform* Langlands–Deligne local constant- External links :...
applied Beck's theorem to Tannakian category
Tannakian category
In mathematics, a tannakian category is a particular kind of monoidal category C, equipped with some extra structure relative to a given field K. The role of such categories C is to approximate, in some sense, the category of linear representations of an algebraic group G defined over K...
theory, greatly simplifying the basic developments.