The Pro-Étale Topos as a Category of Pyknotic Presheaves

Abstract

Let $X$ be a coherent scheme and let $\mathrm{Gal}(X)$ denote the Galois category of $X$, as introduced by Barwick, Glasman and Haine. In this paper we prove that the hypercomplete pro-étale $\infty$-topos $X_{\mathrm{proet}}$ of $X$, introduced by Bhatt and Scholze, is equivalent to the category of continuous representations of $\mathrm{Gal}(X)$ with values in the $\infty$-category of pyknotic spaces.

Final version, to appear in Documenta Mathematica. Theorem 3.11 now explicitly describes the functors in both directions; many other minor changes to improve readability