Etale homotopy theory of non-archimedean analytic spaces

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 étale realization functor of Isaksen. We apply these ideas to define an étale homotopy type functor $\acute{\mathrm{e}}\mathrm{t}(\mathcal{X})$ 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 étale homotopy types coming from Tate's rigid spaces, Huber's adic spaces, and rigid models when they are all defined.

35 pages, comments welcome

References in corpus (1)

• Higher Topos Theory