The fundamental progroupoid of a general topos

TL;DR

This paper extends the concept of the fundamental progroupoid from locally connected to arbitrary topoi, introducing a new definition of covering projections and exploring their classification via localic progroupoids.

Contribution

It generalizes the fundamental progroupoid concept to all topoi and defines covering projections in this broader context, which was not previously established.

Findings

The fundamental progroupoid is a localic progroupoid for any topos.

Not all locally constant objects are covering projections in general topoi.

The classifying topos is no longer a Galois topos in this setting.

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.