Beck's Monadicity Theorem
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In
category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, a branch of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 tripleability theorem because of the older term ''triple'' for a monad. Beck's monadicity theorem asserts that 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 ...
:U: C \to D is monadic if and only if # ''U'' has a left adjoint; # ''U'' reflects
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
s (if ''U''(''f'') is an isomorphism then so is ''f''); and # ''C'' has
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 the ...
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. There are several variations of Beck's theorem: if ''U'' has a left adjoint then any of the following conditions ensure that ''U'' is monadic: *''U'' reflects
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
s and ''C'' has
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 the ...
s of reflexive pairs (those with a common right inverse) and ''U'' preserves those coequalizers. (This gives the crude monadicity theorem.) *Every diagram 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. Another variation of Beck's theorem characterizes strictly monadic functors: those for which the comparison functor is an isomorphism rather than just an
equivalence of categories In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two Category (mathematics), categories that establishes that these categories are "essentially the same". There are numerous examples of cate ...
. For this version the definitions of what it means to create coequalizers is changed slightly: the coequalizer has to be unique rather than just unique up to isomorphism. Beck's theorem is particularly important in its relation with the descent theory, which plays a role in
sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics) In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open s ...
and stack theory, as well as in the
Alexander Grothendieck Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...
's approach to
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
. Most cases of faithfully flat descent of
algebraic structure In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
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 Jean Bénabou and Jacques Roubaud) to be equivalent (under some conditions) to the comonad approach. In a later work,
Pierre Deligne Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoor ...
applied Beck's theorem to Tannakian category theory, greatly simplifying the basic developments.


Examples

*The
forgetful functor In mathematics, more specifically 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 mapping to the output. For an algebraic structure of ...
from topological spaces to sets is not monadic as it does not reflect isomorphisms: continuous bijections between (non-compact or non-Hausdorff) topological spaces need not be homeomorphisms. * shows that the functor from commutative C*-algebras to sets sending such an algebra ''A'' to the
unit ball Unit may refer to: General measurement * Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law **International System of Units (SI), modern form of the metric system **English units, histo ...
, i.e., the set \, is monadic. Negrepontis also deduces Gelfand duality, i.e., the equivalence of categories between the opposite category of compact Hausdorff spaces and commutative C*-algebras can be deduced from this. *The powerset functor from Setop to Set is monadic, where Set is the
category of sets In the mathematical field of category theory, the category of sets, denoted by Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the functions from ''A'' to ''B'', and the composition of mor ...
. More generally Beck's theorem can be used to show that the powerset functor from Top to T is monadic for any topos T, which in turn is used to show that the topos T has finite colimits. *The forgetful functor from
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...
s to sets is monadic. This functor does not preserve arbitrary coequalizers, showing that some restriction on the coequalizers in Beck's theorem is necessary if one wants to have conditions that are necessary and sufficient. *If ''B'' is a faithfully flat
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
over the commutative ring ''A'', then the functor ''T'' from ''A'' modules to ''B'' modules taking ''M'' to ''B''⊗''A''''M'' is comonadic. This follows from the dual of Beck's theorem, as the condition that ''B'' is flat implies that ''T'' preserves finite limits, while the condition that ''B'' is faithfully flat implies that ''T'' reflects isomorphisms. A coalgebra over ''T'' turns out to be essentially a ''B''-module with descent data, so the fact that ''T'' is comonadic is equivalent to the main theorem of faithfully flat descent, saying that ''B''-modules with descent are equivalent to ''A''-modules.


External links

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References

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* * * * * * * * * (3 volumes). * * {{refend Adjoint functors Category theory