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Nonabelian $H^1$ and the Étale Van Kampen Theorem |
Canad. J. Math. 63(2011), 1388-1415
Published:2011-05-13
Printed: Dec 2011
Printed: Dec 2011
- Michael D. Misamore,
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.
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 |
| MSC Classifications: |
18G30 - Simplicial sets, simplicial objects (in a category) [See also 55U10] 14F35 - Homotopy theory; fundamental groups [See also 14H30] |