The author introduced and employed certain `fundamental pushout toposes'
in the construction of the coverings fundamental groupoid of a locally
connected topos. Our main purpose in this paper is to generalize this
construction without the local connectedness assumption. We replace
connected components by constructively complemented, or definable,
monomorphisms. Unlike the locally connected case, where the fundamental
groupoid is localic prodiscrete and its classifying topos is a Galois
topos, in the general case our version of the fundamental groupoid is a
locally discrete progroupoid and there is no intrinsic Galois theory in
the sense of Janelidze. We also discuss covering projections, locally
trivial, and branched coverings without local connectedness by analogy
with, but also necessarily departing from, the locally connected case.
Throughout, we work abstractly in a setting given axiomatically by a
category **V** of locally discrete locales that has as examples the
categories **D** of discrete locales, and **Z** of zero-dimensional
locales. In this fashion we are led to give unified and often simpler
proofs of old theorems in the locally connected case, as well as new ones
without that assumption.

Keywords: topos, fundamental progroupoid, spreads, zero dimensional locales, covering projections

2000 MSC: 18B25, 57M12, 18C15, 06E15

*Theory and Applications of Categories,*
Vol. 20, 2008,
No. 9, pp 186-214.

http://www.tac.mta.ca/tac/volumes/20/9/20-09.dvi

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http://www.tac.mta.ca/tac/volumes/20/9/20-09.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/20/9/20-09.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/20/9/20-09.ps