The Pro-\'Etale Topos as a Category of Pyknotic Presheaves

Abstract

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