Beck's monadicity theorem
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In category theory, a branch of mathematics, 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 coequalizer pairs, and U preserves those coequalizers.
This is a basic result of J. M. 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.
One place this theorem is important is in relation with descent, in particular in the Grothendieck approach to algebraic geometry. Most cases of faithfully flat descent of algebraic structures (e.g. those 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 whole Grothendieck approach via descent data was shown (Benabou and others) to be equivalent to the comonad approach. In later work, Deligne applied Beck's theorem to Tannakian category theory, greatly simplifying the basic developments.
