Extended étale homotopy groups from profinite Galois categories

Abstract

In this note we show that the protruncated shape of a spectral $\infty$-topos is a delocalization of its profinite stratified shape. This gives a way to reconstruct the extended étale homotopy groups (i.e., the non-profinitely complete étale homotopy groups) of a coherent scheme from its profinite Galois category.

9 pages. Comments very welcome. v2: Removed an erroneous remark about the protruncated shape preserving finite products