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

TL;DR

This paper establishes an equivalence between the hypercomplete pro-étale topos of a coherent scheme and the category of continuous representations of its Galois category valued in pyknotic spaces, linking geometric and categorical perspectives.

Contribution

It proves that the hypercomplete pro-étale topos is equivalent to continuous Galois representations in pyknotic spaces, connecting topos theory with pyknotic and Galois categories.

Findings

Hypercomplete pro-étale topos is equivalent to continuous Galois representations in pyknotic spaces.

Establishes a new categorical equivalence linking topos theory and Galois categories.

Provides a framework for understanding pro-étale topoi via 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.