Abstract view
Published:2011-09-15
Printed: Dec 2011
J. F. Jardine,
Mathematics Department, University of Western Ontario, London, ON N6A 5B7
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".