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# Pointed Torsors

Published:2011-09-15
Printed: Dec 2011
• J. F. Jardine,
Mathematics Department, University of Western Ontario, London, ON N6A 5B7
 Format: LaTeX MathJax

## Abstract

This paper gives a characterization of homotopy fibres of inverse image maps on groupoids of torsors that are induced by geometric morphisms, in terms of both pointed torsors and pointed cocycles, suitably defined. Cocycle techniques are used to give a complete description of such fibres, when the underlying geometric morphism is the canonical stalk on the classifying topos of a profinite group $G$. If the torsors in question are defined with respect to a constant group $H$, then the path components of the fibre can be identified with the set of continuous maps from the profinite group $G$ to the group $H$. More generally, when $H$ is not constant, this set of path components is the set of continuous maps from a pro-object in sheaves of groupoids to $H$, which pro-object can be viewed as a Grothendieck fundamental groupoid".
 Keywords: pointed torsors, pointed cocycles, homotopy fibres
 MSC Classifications: 18G50 - Nonabelian homological algebra 14F35 - Homotopy theory; fundamental groups [See also 14H30] 55B30 - unknown classification 55B30

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