The étale and pro-étale exodromy theoremsMPIM

by Remy van Dobben de Bruyn (MPIM)

Tuesday Dec 2, 2025, 2:45 PM → 4:00 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

Minicourse on the étale and pro-étale exodromy theorems

Étale sheaves on the spectrum of a field can be understood as sets with a continuous action of the Galois group. A similar classification is available for locally constant sheaves on an arbitrary base scheme. In 2018, Barwick–Glasman–Haine proved a far reaching generalisation of this result, classifying constructible étale sheaves on a scheme as continuous representations of a profinite category they call the Galois category. This was then extended to pro-étale sheaves by Sebastian Wolf in 2022.

In this minicourse, I will explain a new and more geometric proof of the étale and pro-étale exodromy theorems, obtained in upcoming joint work with Sebastian Wolf. The key player in our method is the condensed category of points of the étale topos. We give an explicit computation of this condensed category in terms of the w-strictly local schemes constructed in Bhatt–Scholze's work on the pro-étale topology. This generalises the computation of the classical (non-condensed) category of points in SGA 4 in terms of strictly Henselian rings. The pro-étale exodromy theorem follows quickly from our computation, and the étale version is deduced from the pro-étale one.

Prerequisites: besides a basic knowledge of algebraic geometry and category theory, we will assume familiarity with the étale and pro-étale topology on a scheme, and the basics of the theory of condensed sets (we do not need much beyond the first definitions). We will not assume any knowledge of higher category theory, but we will indicate where ∞-categorical statements can be obtained.