Beck's monadicity theorem
In category theory, a branch of mathematics, Beck's monadicity theorem gives a criterion that characterises monadic functors, introduced by Beck (2003) 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
is monadic if and only if
- U has a left adjoint;
- U reflects isomorphisms; and
- C has coequalizers 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 isomorphisms and C has coequalizers of reflexive pairs (those with a common right inverse) and U preserves those coequalizers. (This gives the crude monadicity theorem.)
- 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.
Another variation of Beck's theorem characterizes strictly monadic functors: those for which the comparison functor is an isomorphism rather than just an equivalence. 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 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 structures (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.
- It follows from Beck's theorem that the forgetful functor from compact Hausdorff spaces to sets is monadic. The left adjoint is the Stone–Čech compactification, the forgetful functor preserves all colimits, and it reflects isomorphisms because any continuous bijection from a compact space to a Hausdorff space is a homeomorphism. The forgetful functor from topological spaces to sets is not monadic as it does not reflect isomorphisms.
- The powerset functor from Setop to Set is monadic, where Set is the category of sets. 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 semigroups 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 over the commutative ring A then the functor T from A modules to B modules taking M to B⊗AM is a comonad. This follows from the dual of Becks theorem, as the condition that B is flat implies that T preserves 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 a comonad is equivalent to the main theorem of faithfully flat descent, saying that B-modules with descent are equivalent to A-modules.
- Pedicchio & Tholen (2004) p.228
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