Beck's monadicity theorem

In category theory, a branch of mathematics, Beck's monadicity theorem asserts that a functor

:$U:\; C\; o\; D$

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. Its importance arises in relation with descent, in particular in the Grothendieck approach to algebraic geometry. Passing to a category of coalgebras for a comonad "T" is a high-flown way of modelling what taking equivalence classes does, in less touchy situations. The theorem, often also called the Beck tripleability theorem because of the older term "triple" for comonads, 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, somewhat non-obviously, to the comonad approach. In later work, and after a hiatus of communication between algebraic geometers and category theorists, Deligne applied Beck's theorem to Tannakian category theory, greatly simplifying the basic developments.