Geometric arcs and fundamental groups of rigid spaces

TL;DR

This paper introduces a new concept of geometric coverings for rigid spaces, expanding the class of coverings and establishing a Galois correspondence, bridging ideas from algebraic geometry and topology.

Contribution

It develops the notion of geometric coverings for rigid spaces, generalizing previous work and proving they form a Galois category classified by a topological group.

Findings

Geometric coverings are closed under disjoint unions.

They are étale local on the rigid space.

Form a tame infinite Galois category for connected spaces.

Abstract

We develop the notion of a geometric covering of a rigid space X, which yields a much larger class of covering spaces than that studied previously by de Jong. Geometric coverings of X are closed under disjoint unions and are \'etale local on X. If X is connected, its geometric coverings form a tame infinite Galois category, and hence are classified by a topological group. The definition is based on the property of lifting of "geometric arcs," making it similar to geometric coverings of schemes studied by Bhatt and Scholze as well as semicoverings of topological spaces introduced by Brazas.