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 \'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.