ag.algebraic geometry - Looking for reference talking about relationship between descent theory and cohomological descent

The relationship between cohomological descent and Lurie's Barr-Beck is exactly the same as the relationship between ordinary descent and ordinary Barr-Back. To put things somewhat blithely, let's say you have some category of geometric objects $mathsf{C}$ (e.g. varieties) and some contravariant functor $mathsf{Sh}$ from $mathsf{C}$ to some category of categories (e.g. to $X$ gets associated its derived quasi-coherent sheaves, or as in SGA its bounded constructible complexes of $ell$-adic sheaves). Now let's say you have a map $p:Y rightarrow X$ in $mathsf{C}$, and you want to know if it's good for descent or not. All you do is apply Barr-Beck to the pullback map $p^ast:mathsf{Sh}(X)rightarrow mathsf{Sh}(Y)$. For this there are two steps: check the conditions, then interpret the conclusion. The first step is very simple -- you need something like $p^ast$ conservative, which usually happens when $p$ is suitably surjective, and some more technical condition which I think is usually good if $p$ isn't like infinite-dimensional or something, maybe. For the second step, you need to relate the endofunctor $p^ast p_ast$ (here $p_*$ is right adjoint to $p^ast$... you should assume this exists) to something more geometric; this is possible whenever you have a base-change result for the fiber square gotten from the two maps $p:Y rightarrow X$ and $p:Y rightarrow X$ (which are the same map). For instance in the $ell$-adic setting you're OK if $p$ is either proper or smooth (or flat, actually, I think). Anyway, when you have this base-change result (maybe for p as well as for its iterated fiber products), you can (presumably) successfully identify the algebras over the monad $p^ast p_ast$ (should I say co- everywhere?) with the limit of $mathsf{Sh}$ over the usual simplicial object associated to $p$, and so Barr-Beck tells you that $mathsf{Sh}(Y)$ identifies with this too, and that's descent. The big difference between this homotopical version and the classical one is that you need the whole simplicial object and not just its first few terms, to have the space to patch your higher gluing hopotopies together.

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