Grothendieck Galois theory and \'etale exodromy

TL;DR

This paper provides a simplified proof of the étale exodromy theorem for constructible sheaves, connecting it to Grothendieck's Galois theory and streamlining previous complex approaches.

Contribution

It offers a quick, Grothendieck-style proof of étale exodromy for constructible sheaves, simplifying the existing long and technical proofs.

Findings

Simplified proof of étale exodromy theorem

Connection between exodromy and Galois theory

Streamlined approach reduces complexity

Abstract

Finite \'etale covers of a connected scheme $X$ are parametrised by the \'etale fundamental group via the monodromy correspondence. This was generalised to an exodromy correspondence for constructible sheaves, first in the topological setting by MacPherson, Treumann, Lurie, and others, and recently also for \'etale and pro-\'etale sheaves by Barwick--Glasman--Haine and Wolf. The proof of the \'etale exodromy theorem is long and technical, using many new definitions and constructions in the $(\infty,2)$-category of $\infty$-topoi. This paper gives a quick proof of the \'etale exodromy theorem for constructible sheaves of sets (with respect to a fixed stratification), in the style of Grothendieck's Galois theory.