Amnon Yekutieli, BGU

Higher Descent

Abstract: Classical descent is about gluing a global geometric object out of local information. Or conversely, it is about classifying global geometric objects using open coverings and cocycles.

I will begin the lecture with a rather thorough discussion of how descent theory let's us classify twisted forms of a sheaf on a topological space (this is 1st nonabelian cohomology).

Next I will recast this geometric construction in terms of cosimplicial groups, in this way getting something that is of purely combinatorial nature.

Higher descent refers to the classification of twisted forms of a stack on a topological space (a sort of 2nd nonabelian cohomology). But in this talk I will adhere to the combinatorial point of view, so stacks will only appear as motivations, and in one or two examples. Thus, in the talk, ``higher descent'' will be mostly a study of cosimplicial crossed groupoids and their descent data. (These concepts will be defined.)

I will present a recent result, the ``Equivalence Theorem'', and mention its role in my work on twisted deformation quantization. Finally I will briefly discuss how model structures enter the picture, and a very new proof of this theorem by Prezma.

There will be a few pictures.

- Lecture notes are at http://www.math.bgu.ac.il/~amyekut/lectures/higher-descent/notes.pdf
- The paper is at http://arxiv.org/abs/1109.1919

(Dec 2011)