Etale homotopy theory of non-archimedean analytic spaces

TL;DR

This paper develops a new framework for étale homotopy theory of non-archimedean analytic spaces, connecting shape theory, cohomology, and various models of these spaces.

Contribution

It introduces a new étale homotopy type functor for Berkovich spaces and compares it across different models, extending existing realization functors.

Findings

Defined an étale homotopy type functor for Berkovich spaces

Extended Isaksen's étale realization functor via a new localization of profinite spaces

Compared étale homotopy types from Tate's, Huber's, and rigid models

Abstract

We review the shape theory of $\infty$-topoi, and relate it with the usual cohomology of locally constant sheaves. Additionally, a new localization of profinite spaces is defined which allows us to extend the \'etale realization functor of Isaksen. We apply these ideas to define an \'{e}tale homotopy type functor $\mathrm{\acute{e}t}(\mathcal{X})$ for Berkovich's non-archimedean analytic spaces $\mathcal{X}$ over a complete non-archimedean field $K$ and prove some properties of the construction. We compare the \'etale homotopy types coming from Tate's rigid spaces, Huber's adic spaces, and rigid models when they are all defined.