The fundamental progroupoid of a general topos

Eduardo J. Dubuc

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arXiv · 0706.1771

The fundamental progroupoid of a general topos

Eduardo J. Dubuc

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Abstract

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 \emph{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.

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BibTeX RIS

Eduardo J. Dubuc. 2007-06-12. The fundamental progroupoid of a general topos. https://arxiv.org/abs/0706.1771

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