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

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.