Atlas: Galois toposes and complete spreads by Marta Bunge

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IV Iberoamerican Conference on Topology and its Applications (IV CITA)
April 18-21, 2001
University of Coimbra
Coimbra, Portugal

Organizers
Maria Manuel Clementino, Jorge Picado, Lurdes Sousa, Maria João Ferreira, Gonçalo Gutierres, Dirk Hofmann

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Galois toposes and complete spreads
by
Marta Bunge
McGill University

A Galois topos is any pointed connected atomic Grothendieck topos E that is generated by its normal atoms (A. Grothendieck, SGA1, LNM 224, 1971). Galois toposes are characterized as those Grothendieck toposes equivalent to the classifying topos BG of a prodiscrete localic group G (I. Moerdijk, ``Prodiscrete groups and Galois toposes", Proc. Kon. Ned. Akad. van Wetenschappen, A-92 (1989) 219-234). Equivalently, a Grothendieck topos is a (pointed) Galois topos iff it is pointed, connected locally connected, and generated by its locally constant coverings. The notion of a Galois topos is meaningful also in the unpointed case.

Theorem 1. Let E be a Grothendieck topos. Then the following are equivalent:

E is an unpointed Galois topos.

E is equivalent to the classifying topos BG for G a prodiscrete localic groupoid.

E is connected locally connected and the canonical geometric morphism E --> \Pi₁^cov(E) is an equivalence, where \Pi₁^cov(E) is the Grothendieck, or coverings fundamental groupoid topos of (the unpointed topos) E (cf. M. Bunge, ``Classifying toposes and fundamental localic groupoids", CT'91, Canad. Math. Soc. Conf. Proc. 13, 1992, 75-96).

An object X in a connected locally connected topos E is said to be a complete spread object if the local homeomorphism E/X --> E is a complete spread (M. Bunge and J. Funk, ``Spreads and the Symmetric Topos II", J. Pure Appl. Alg. 130 (1998) 49-84). Denote by \Pi₁^cso(E) the full subcategory of E generated by its complete spread objects. Recall that \Pi₁^path(E) denotes the paths fundamental groupoid topos of E (cf. I. Moerdijk and G. Wraith, ``Connected locally connected toposes are path connected", Trans. AMS 295 (1986) 849-859) and that there is a ``comparison map" \Pi₁^path(E) --> \Pi₁^cov(E) (M. Bunge and I. Moerdik, ``On the construction of the Grothendieck fundamental group of a topos by paths", J. Pure Appl. Alg. 116 (1997) 99-113) which is an equivalence under the additional assumption that E is locally paths simply connected, just like in topology.

The proof of the following theorem is based on results obtained in M. Bunge, ``Universal Covering Localic Toposes", Comptes Rendues Acad. Sci. Canada 14 (1992) 245-250, and in M. Bunge and J. Funk, ``Spreads and the Symmetric Topos II", J. Pure Appl. Alg. 130 (1998) 49-84.

Theorem 2. Let E be a connected locally connected Grothendieck topos. Then there exist geometric morphisms \Pi₁^path(E) --> \Pi₁^cso(E) --> \Pi₁^cov(E).

In general, these geometric morphisms need not be equivalences, but if E is either locally paths simply connected or a Galois topos, then all three toposes above are equivalent.

Partial results have also been obtained concerning a possible characterization of toposes of the form \Pi₁^cso(E), in connection with the classifying toposes B(G) for G a totally disconnected localic groupoid. We note that for locales, not every totally disconnected locale need be discrete (unless it is locally connected).

http://www.math.mcgill.ca/~bunge

Date received: February 25, 2001