Abstract
Abstract Generalized étale homotopy pro-groups associated with pointed, connected, small Grothendieck sites 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 , a description of Grothendieck's short exact sequence for Galois descent in terms of pointed torsor trivializations, and a new étale van Kampen theorem that gives a simple statement about a pushout square of pro-groups that works for covering families that do not necessarily consist exclusively of monomorphisms. A corresponding van Kampen result for Grothendieck's profinite groups immediately follows.
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