The fundamental progroupoid of a general topos.

EDUARDO J. DUBUC

JOURNAL OF PURE AND APPLIED ALGEBRA

Elsevier

Lugar: Amsterdam; Año: 2008 vol. 212 p. 2479 - 2492

0022-4049

  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 is no longer a Galoistopos. 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.