Geometry and Topology Seminar, 8 NOV, Nicholas Meadows. Title “Quasi-Categories and Simpson’s Theory of Higher Stacks”

Department of Mathematics                                                           University of  Haifa

                                                   Geometry & Topology Seminar

Speaker:  Nicholas Meadows(University of Haifa)

Topic: Quasi-Categories and Simpson’s Theory of Higher Stacks

Place:    room 614 in the Science & Education Building

Time   12:10

Date:  Thursday, November 8, 2018


Abstract: In their 2001 preprint, `Descent Pour Les N-Champs’ Hirschowitz and Simpson established a theory of $(\infty, n)$-stacks for all integers n. Their work has had a large amount of influence on the Lurie and Toen approaches to derived algebraic geometry. The wider goal of Simpson’s research program was to develop tools to study non-abelian Hodge theory.

In my thesis, I developed a model structure on simplicial presheaves on a small site in which the weak equivalences are maps that induce Joyal weak equivalences on stalks (the local Joyal model structure). The purpose of this talk is to explain how to reininterpret Simpson’s results on $(\infty, 1)$-stacks using the Joyal and local Joyal model structure, which give a much more tractable presentation of the results.
The results covered include a characterization of higher stacks in terms of mapping space presheaves, and a general technique for constructing higher stacks from presheaves of simplicial model categories on a site. This leads to the construction of the higher stack of simplicial $\mathcal{R}$-module spectra, where $\mathcal{R}$ is a sheaf of rings on the site. This object is related to the problem of glueing together unbounded chain complexes along quasi-isomorphisms