I am now taking a course focusing on triangulated geometry. The professor has formulated the Beck's theorem for Karoubian triangulated category. The proof is very simple. Just using the universal homological functor(equivalent to Verdier abelianization)back to abelian settings(in particular, Frobenius abelian category), then using the Beck's theorem for abelian category to get the proof.

When he finished the proof, he made a remark that the cohomological descent theory can be taken as a consequence of triangulated version of Beck's theorem.

As we know, the Beck's theorem for abelian category is equivalent to Grothendieck flat descent theory(Beck's theorem may be more general). I have two questions:

Is there any reference(other than SGA 4)in English explaining the relationship of usual descent theory and cohomolgical descent theory? What I am looking for is not a very thick book but a lecture notes with some examples.

I know Jacob Lurie developed the derived version of Beck's theorem for his infinity category(correct me if I make mistake). But I have never read his paper very carefully. I wonder whether he explained the relationship of Beck's theorem and "cohomogical descent in his settings"(if there exists such terminology).

All the other comments are welcome. Thank you