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

  Published:2011-05-13
 Printed: Dec 2011
  • Michael D. Misamore,
    Universität Duisburg-Essen, Universitätsstr. 2, 45141 Essen, Germany
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Abstract

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 étale homotopy theory, simplicial sheaves
MSC Classifications: 18G30, 14F35 show english descriptions Simplicial sets, simplicial objects (in a category) [See also 55U10]
Homotopy theory; fundamental groups [See also 14H30]
18G30 - Simplicial sets, simplicial objects (in a category) [See also 55U10]
14F35 - Homotopy theory; fundamental groups [See also 14H30]
 

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