2007/06/12 by Eduardo J. Dubuc, Dubuc, Eduardo J.
Mathematics · #18B25 #18F99 #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #math.AT #math.CT #msc:18B25 #msc:18F99
paper · pdf · doi:10.48550/arxiv.0706.1771
19 pages
arxiv created 2007/06/12 · arxiv updated 2009/12/01
It is well known that the category of covering projections (that is, locally constant objects) of a locally connected topos is equivalent to the classifying topos of a strict progroupoid (or, equivalently, a localic prodiscrete groupoid), the fundamental progroupoid, and that this progroupoid represents first degree cohomology. In this paper we generalize these results to an arbitrary topos. The fundamental progroupoid is now a localic progroupoid, and can not be replaced by a localic groupoid. The classifying topos in not any more a Galois topos. Not all locally constant objects can be considered as covering projections. The key contribution of this paper is a novel definition of covering projection for a general topos, which coincides with the usual definition when the topos is locally connected. The results in this paper were presented in a talk at the Category Theory Conference, Vancouver July 2004.