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1. CJM 2011 (vol 63 pp. 1388)

Misamore, Michael D.
Nonabelian $H^1$ and the Étale Van Kampen Theorem
Generalized étale homotopy pro-groups $\pi_1^{\operatorname{ét}}(ċ{C}, x)$ associated with pointed, connected, small Grothendieck sites $(\mathcal{C}, x)$ 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 $\pi_1^{\operatorname{ét}}(ét(X), x)$, a description of Grothendieck's short exact sequence for Galois descent in terms of pointed torsor trivializations, and a new étale van Kampen theorem 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 $\pi_1^{\mathrm{Gal}}$ immediately follows.

Keywords:étale homotopy theory, simplicial sheaves
Categories:18G30, 14F35

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