I am resuming these days my work on descent theory. In August/September I was thinking on issues related to the descent for
equivariant functors between categories where each category is itself glued from pieces. Equivariant in the sense of coactions of comonads or a similar formalism. One of the motivations is study of principal bundles over noncommutative schemes. The Cech cocycles are a bit tricky here in full generality, there are phenomena which do not exist or do not matter in commutative context.
I will meet these days Gabi Bohm from Budapest to talk about such issues; she is well versed in comonads and related issues. This will be also excuse to post here some standard and not so standard background from selected parts of descent theory. This post will grow in few days. Keep tuned 
