group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
symmetric monoidal (∞,1)-category of spectra
Higher monadic descent is the generalization of the notion of monadic descent from category theory to higher category theory. It relates to the descent of ∞-stacks as ordinary monad descent relates to stacks.
See also cohomological descent.
For $\phi \colon B \longrightarrow A$ a homomorphism of suitable monoids, there is the corresponding pull-push adjunction (extension of scalars $\dashv$ restriction of scalars) on categories of modules
The bar construction of the corresponding monad is the corresponding Amitsur complex.
(e.g. Hess 10, section 6)
A monadicity theorem for pseudomonads is proved in
In 2-category-theory, a notion of 2-fibered categories are defined in Gray’s work and then again introduced and discussed at length by Claudio Hermida, who has good ideas on higher n (and I will be trilled to hear once that he found the time to return to the topic and give us good answers). An appendix in
is discussing a 2-categorical version of Beck-Chevalley condition needed to compare it with the 2-monadic descent; there is also a version of Beck’s theorem sketched there.
A comprehensive treatment in the context of (∞,1)-category-theory, general theory of (∞,1)-monads and their monadicity theorem is in
later absorned as
Unfortunately, Kontsevich’s monadicity theorem (July 2004) which is in the setup of A-∞-categories, still remains unpublished. The triangulated version is in Rosenberg’s lectures
Its proof is based on Verdier's abelianization functor.
See also, for another point of view,
Discussion of monadic descent for simplicially enriched categories is in
and for quasi-categories in