Nonabelian $H^1$ and the Étale van Kampen Theorem

英文摘要：

Generalized étale homotopy pro-groups $π_1^{\ets}(\mc{C}, x)$ 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 $π_1^{\ets}(\et{X}, x)$, a description of Grothendieck's short exact sequence for Galois descent in terms of pointed torsor trivializations, and a new étale 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 $π_1^{\Gals}$ immediately follows.