group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The theory of cohomological descent deals with the question if the derived analogue of the (co)monadic comparison functor is fully faithful (or more rarely an equivalence of categories) when formulated at the level of total derived functors and derived categories, and usually taken with respect to hypercovers.
The notion has been introduced in
A summary is also in
CRM Proc. Lecture Notes, 24, Amer. Math. Soc., Providence, RI, 2000.
For a readable introduction see
Closely related is the monadic descent in triangulated context in the sense of page 36-37 in
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