The goal of this seminar is to understand the work of Barwick, Glasman, and Haine on stratified étale homotopy theory, following their paper/book Exodromy [Exo]. This is a generalisation of monodromy representations of ℓ-adic local systems to constructible sheaves.
We will start with the version for stratified topological spaces, where MacPherson defined the exit path category, an analogue of the fundamental groupoid for stratified spaces. Next, we will discuss higher categorical versions due to Treumann and Lurie. We then spend some time reviewing classical étale homotopy theory, and proceed to the stratified étale homotopy type.
The ultimate goal is to get to the exodromy theorem for ℓ-adic constructible sheaves. As in the monodromy case, there is a topology that interacts with the ℓ-adic topology. To make sense of this in a higher categorical context, we need to talk about pyknotic objects, which some of you may know under the different guise of condensed mathematics.
The format is loosely inspired by Akshay's Thursday seminar. We aim for informal talks, and we can take as much time as we need. The seminar should be accessible to graduate students and postdocs, so we plan to spend substantial time on the basics of simplicial homotopy theory and higher category theory (in fact this is one of the reasons for picking this topic).
Talks will be on Tuesdays 1:30–3:00 (ET) through Zoom. A handful of IAS participants will be able to attend in person in S-101, although the speaker (and organiser) will not always be there. There are two ways to attend in person: register in advance by emailing me (Remy), or first come first served.
We will start with an overview of the seminar and a discussion of MacPherson's result for stratified topological spaces. Very roughly, the goal of the first semester is to learn enough about simplicial sets, higher category theory, and higher topos theory in order to be able to read [Exo].
In the second semester, we will dive into the work of Barwick, Glasman, and Haine, leading up to a general exodromy result for ∞-topoi stratified over a finite poset or spectral topological space. We then turn to artihmetic aspects: étale homotopy theory, pyknotic and condensed mathematics, and ultimately the étale version of the exodromy correspondence.
The tentative plan outlined here is subject to change, and will be updated regularly. It is similar but not identical to that of the Heidelberg Oberseminar on the same topic (summer 2019).
The main references are:
Note that Treumann also has a thesis with the same title, which follows different numbering and contains more material.
Because of the new member talks, we will not start until the first full week of October.
The schedule below will be updated regularly, and I'm open to suggestions!