Exodromy

TL;DR

This paper introduces a topological category called l(X) that encodes the étale topology of a scheme, establishing an equivalence with constructible sheaves and providing new insights into the scheme's étale homotopy type and topos reconstruction.

Contribution

It constructs the l(X) category as an étale exit-path category, extending MacPherson's concept, and proves an exodromy equivalence with various coefficients, including nonabelian and profinite modules.

Findings

Establishes an equivalence between representations of l(X) and constructible sheaves.

Provides a new description of the étale homotopy type of schemes.

Reconstructs the étale topos from l(X) and proves a higher categorical Hochster Duality.

Abstract

Let $X$ be a quasicompact quasiseparated scheme. Write $\operatorname{Gal}(X)$ for the category whose objects are geometric points of $X$ and whose morphisms are specializations in the \'etale topology. We define a natural profinite topology on the category $\operatorname{Gal}(X)$ that globalizes the topologies of the absolute Galois groups of the residue fields of the points of $X$. One of the main results of this book is that $\operatorname{Gal}(X)$ variant of MacPherson's exit-path category suitable for the \'etale topology: we construct an equivalence between representations of $\operatorname{Gal}(X)$ and constructible sheaves on $X$. We show that this 'exodromy equivalence' holds with nonabelian coefficients and with finite abelian coefficients. More generally, by using the pyknotic/condensed formalism, we extend this equivalence to coefficients in the category of modules over profinite rings and algebraic extensions of $\mathbf{Q}_{\ell}$. As an 'exit-path category', the topological category $\operatorname{Gal}(X)$ also gives rise to a new, concrete description of the \'etale homotopy type of $X$. We also prove a higher categorical form of Hochster Duality, which reconstructs the entire \'etale topos of a quasicompact and quasiseparated scheme from the topological category $\operatorname{Gal}(X)$. Appealing to Voevodsky's proof of a conjecture of Grothendieck, we prove the following reconstruction theorem for normal varieties over a finitely generated field $k$ of characteristic $0$: the functor $X\mapsto\operatorname{Gal}(X)$ from normal $ k $-varieties to topological categories with an action of $\operatorname{G}_{k}$ and equivariant functors that preserve minimal objects is fully faithful.