Variants of the de Jong fundamental group

TL;DR

This paper clarifies the structure of certain covering categories of rigid spaces, showing their differences from de Jong's original definitions and establishing their relation to Scholze's pro-étale topology.

Contribution

It demonstrates that the category of admissible coverings differs from de Jong's but still forms a tame Galois category, and characterizes étale coverings as locally constant sheaves in the pro-étale topology.

Findings

The category of admissible coverings is different from de Jong's original category.

It still forms a tame infinite Galois category.

Objects of the étale covering category correspond to locally constant sheaves in the pro-étale topology.

Abstract

For a rigid space $X$, we answer two questions of de Jong about the category $\mathbf{Cov}^\mathrm{adm}_X$ of coverings which are locally in the admissible topology on $X$ the disjoint union of finite etale coverings: we show that this class is different from the one used by de Jong, but still gives a tame infinite Galois category. In addition, we prove that the objects of $\mathbf{Cov}^\mathrm{et}_X$ (with the analogous definition) correspond precisely to locally constant sheaves for the pro-etale topology defined by Scholze.