Nonabelian $H^1$ and the \'Etale van Kampen Theorem

TL;DR

This paper introduces generalized étale homotopy pro-groups, explores their connections to Galois theory and torsors, and presents a new étale van Kampen theorem applicable to broader covering families.

Contribution

It defines generalized étale homotopy pro-groups and establishes a new van Kampen theorem that extends previous results to more general covering families.

Findings

Provides rigorous proofs of folklore results about étale fundamental groups.

Describes Grothendieck's short exact sequence in terms of torsor trivializations.

Introduces a van Kampen theorem for pro-groups applicable to non-monomorphic coverings.

Abstract

Generalized \'etale homotopy pro-groups $\pi_1^{\ets}(\mc{C}, x)$ associated to pointed connected small Grothendieck sites $(\mc{C}, x)$ are defined and their relationship to Galois theory and the theory of pointed torsors for discrete groups is explained. Applications include new rigorous proofs of some folklore results around $\pi_1^{\ets}(\et{X}, x)$, a description of Grothendieck's short exact sequence for Galois descent in terms of pointed torsor trivializations, and a new \'etale van Kampen theorem which gives a simple statement about a pushout square of pro-groups that works for covering families which do not necessarily consist exclusively of monomorphisms. A corresponding van Kampen result for Grothendieck's profinite groups $\pi_1^{\Gals}$ immediately follows.