Algebraic Geometry Seminar

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Fun with Gerbes, Part 3

Speaker: Michael McQuillan, IHES

Location: Warren Weaver Hall 317

Date: Tuesday, May 12, 2015, 5 p.m.

Synopsis:

This will be the last in the series of "Fun with gerbes" talks, and will largely require no knowledge of the previous two. The theme of these talks has been étale homotopy 2-types via Grothendieck's theory of champs (the translation stack is awful), and this final talk will concentrate on applications. A particular feature of having the right definition of higher homotopy groups via n-categories is that the Lefschetz theorem becomes a more or less tautological induction. At the very least I'll explain the initial π₀ step in the induction -- typically all problems in this area are, when correctly understood, problems about π₀. There are other fun applications like the failure of the smooth base change theorem for étale homotopy even though it's a theorem of étale cohomology, but this is sort of disjoint from the Lefschetz theorems, so, if time allows for doing anything more than Lefschetz for π₀, we can then vote on whether to look at Lefschetz for the higher homotopy groups, or the failure of the smooth base change theorem. A manuscript containing all the material from the "Fun with gerbes" series ought to be available by the end of the week.