# Extended \'etale homotopy groups from profinite Galois categories

## TL;DR

This paper introduces a method to reconstruct extended étale homotopy groups of coherent schemes using their profinite Galois categories, linking spectral topos theory with algebraic geometry.

## Contribution

It establishes a connection between protruncated shapes of spectral ∞-topoi and profinite stratified shapes, enabling the reconstruction of non-profinite étale homotopy groups.

## Key findings

- Protruncated shape is a delocalization of profinite stratified shape.
- Extended étale homotopy groups can be reconstructed from profinite Galois categories.
- Provides a new perspective linking spectral topos theory with algebraic geometry.

## 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 \'etale homotopy groups (i.e., the non-profinitely complete \'etale homotopy groups) of a coherent scheme from its profinite Galois category.