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Classifying toposes and fundamental localic groupoids. (English) Zbl 0787.18005

Seely, R. A. G. (ed.), Category theory 1991. Proceedings of an international summer category theory meeting, held in Montréal, Québec, Canada, June 23-30, 1991. Providence, RI: American Mathematical Society. CMS Conf. Proc. 13, 75-96 (1992).
The notion of a totally disconnected topos is introduced. If \(\mathcal E\) is a topos which is bounded over a base topos \(\mathcal J\), and if \(\mathcal E\) admits a locale of path components in \(\mathcal J\), the notion of a totally disconnected topos is used to define the fundamental localic groupoid of \(\mathcal E\). This fundamental localic groupoid is a totally disconnected open étale complete localic groupoid in \(\mathcal J\), and it represents the first degree cohomology of \(\mathcal E\) with coefficients in discrete groups. The relation with the fundamental group of a pointed Grothendieck topos is also exhibited: on an open cover of the base topos \(\mathcal J\), the fundamental localic groupoid of \(\mathcal E\) is locally Morita equivalent to a prodiscrete localic groupoid.
For the entire collection see [Zbl 0771.00047].

MSC:

18B25 Topoi
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
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