It is well known that Beck's theorem for Comonad is equivalent to Grothendieck flat descent theory on scheme.
There are several version of derived noncommutative geometry. I wonder whether someone developed the triangulated version of Beck's theorem. And What does it mean,if exists?
There isn't a descent theory for derived categories per se - one can't glue objects in the derived category of a cover together to define an object in the base. (Trying to apply the usual Barr-Beck to the underlying plain category doesn't help.)
But I think the right answer to your question is to use an enriched version of triangulated categories (differential graded or $A_\infty$ or stable $\infty$-categories), for which there is a beautiful Barr-Beck and descent theory, due to Jacob Lurie. (This is discussed at length in the n-lab I believe, and came up recently on the n-category cafe (where I wrote basically the same comment here..) This is proved in DAG II: Noncommutative algebra. In the comonadic form it goes like this. Given an adjunction between $\infty$-categories (let's call the functors pullback and pushforward, to mimic descent), if we have
then the $\infty$-category downstairs is equivalent to comodules over the comonad (pullback of pushforward). (There's an opposite monadic form as well) This can be verified in the usual settings where you expect descent to hold. In other words if you think of derived categories as being refined to $\infty$-categories (which have the derived category as their homotopy category), then everything you might want to hold does.
So while derived categories don't form a sheaf (stack), their refinements do: you can recover a complex (up to quasiisomorphism) from a collection of complexes on a cover, identification on overlaps, coherences on double overlaps, coherences of coherences on triple overlaps etc. More formally: define a sheaf as a presheaf $F$ which has the property that for an open cover $U\to X$, defining a Cech simplicial object $U_\bullet=\{U\times_X U\times_X U\cdots\times_X U\}$, then $F(X)$ is the totalization of the cosimplicial object $F(U_\bullet)$. Then enhanced derived categories form sheaves (in appropriate topologies) as you would expect. This is of course essential to having a good theory of noncommutative algebraic geometry!
There is a Beck's theorem for Karoubian triangulated categories, proposed by Konstevich and Rosenberg in July 2004, which is proved by using the Verdier's abelianization functor and graded monads; see page 36 of A. Rosenberg, Topics in noncommutative algebraic geometry, homological algebra and K-theory, preprint MPIM Bonn 2008-57, pdf.
In fact they got simulatenously (July 15, 2004) both versions: A-infinity and triangulated. The reference to the triangulated is above: while for A-infinity there is no write up, but Kontsevich gave a talk (I think Nov 2004, van den Bergh birthday conference) where he formulated and used the result; one of nice applications was to glue certain ordinary commutative schemes to get certain formal schemes. I remember very well the weeks preceding the result when we discussed possible shape of the result seeked at IHES. Later at the conference in Split, Kontsevich gave a talk where he gave some usages in noncommutative algebraic geometry.
I disagree with the statement: "Beck's theorem for comonad is equivalent to Grothendieck flat descent theory on scheme". Namely Grothendieck gave both the flat descent theory for quasicoherent sheaves (SGA I.8.1) which is a special case of Beck's theorem (though it has some symmetries which general noncommutative case does not have), but also the (stronger) flat descent theorem for affine schemes (SGA I.8.2) and for morphisms (cf. SGA I.8.5), which unlike the descent for quasicoherent sheaves, does not generalize to the noncommutative algebras and consequently to categorical setup either.
