In category theory, a branch of mathematics, Beck's monadicity theorem gives a criterion that characterises monadic functors, introduced by Jonathan Mock Beck (2003) in about 1964.
It is sometimes called the Beck tripleability theorem because of the older term triple for a monad.
Beck's theorem is particularly important in its relation with the descent theory, which plays a role in sheaf and stack theory, as well as in the Alexander Grothendieck's approach to algebraic geometry.
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 applied Beck's theorem to Tannakian category theory, greatly simplifying the basic developments.