The pro-étale topos as a category of pyknotic presheaves

Abstract

In this paper we will show that for a quasicompact quasiseparated scheme $X$ the hypercomplete pro-étale $\infty$-topos, as introduced by Bhatt and Scholze, is equivalent to the $\infty$-category of continuous representations of the Galois category $\mathrm{Gal}(X)$ of $X$ with values in the $\infty$-category of pyknotic spaces. In particular this proves that internally to pyknotic spaces, the hypercomplete pro-étale $\infty$-topos of $X$ is an $\infty$-category of presheaves.