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An application of descent to a classification theorem for toposes

Published online by Cambridge University Press:  24 October 2008

Marta Bunge
Marta Bunge
Affiliation:
Department of Mathematics and Statistics, McGill University, Montréal, P. Québec, CanadaH3A 2K6
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Extract

The aim of this paper is to answer the following question. For a spatial groupoid G, i.e. for a groupoid in the category Sp of spaces (in the sense of [20]) in a topos , and continuous maps, the topos BG, of étale G-spaces, is called ‘the classifying topos of G’ by Moerdijk[22]. This terminology is suggested by the case of G a discrete group (in Sets), as then BG, the topos of G-sets, classifies principal G-bundles. This means that, for each topological space X, there is a bijection between isomorphism classes of principal G-bundles over X and isomorphism classes of geometric morphisms from Sh(X) to BG. The question is: what does BG classify, in terms of G, in the general case of a spatial groupoid G in a topos ?

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 107 , Issue 1 , January 1990 , pp. 59 - 79
Copyright
Copyright © Cambridge Philosophical Society 1990

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References

REFERENCES

[1]Artin, M., Grothendieck, A. and Verdier, J. L.. Théorie des Topos et Cohomologie Étale des Schémas. Lecture Notes in Math. vol. 269 (Springer-Verlag, 1972).Google Scholar
[2]Barr, M.. Exact categories. In Exact Categories and Categories of Sheaves, Lecture Notes in Math. vol. 236 (Springer-Verlag, 1970), pp. 1–120.Google Scholar
[3]Bourn, D.. Distributeurs et Champ associé. Cahiers Topologie Geom. Différentielle Catégoriques 21 (1980), 403–409.Google Scholar
[4]Bourn, D.. The shift functor and the comprehension factorization for internal groupoids. Cahiers Topologie Geom. Différentielle Catégoriques 28 (1987), 197–226.Google Scholar
[5]Bunge, M. and Paré, R.. Stacks and equivalence of indexed categories. Cahiers Topologie Geom. Différentielle Catégoriques 20 (1979), 373–399.Google Scholar
[6]Bunge, M.. Stack completions and Morita equivalences for category objects in a topos. Cahiers Topologie Geom. Différentielle Catégoriques 20 (1979), 401–436.Google Scholar
[7]Diaconescu, R.. Grothendieck toposes have Boolean points – a new proof. Comm. Algebra 4 (1976), 723–729.CrossRefGoogle Scholar
[8]Demazure, M. and Gabriel, P.. Groupes Algebriques, Tome I (Masson & Cie, 1970).Google Scholar
[9]Dubuc, E.. Adjoint triangles. In Reports of the Midwest Category Seminar II, Lecture Notes in Math. vol. 61 (Springer-Verlag, 1968), pp. 69–91.CrossRefGoogle Scholar
[10]Duskin, J.. An outline of non-abelian cohomology in a topos: (1) The theory of bouquets and gerbes. Cahiers Topologie Geom. Différentielle Catégoriques 23 (1982), 165–191.Google Scholar
[11]Duskin, J.. Non-abelian cohomology in a topos. (Manuscript, SUNY at Buffalo.)Google Scholar
[12]Husemoller, D.. Fibre Bundles (McGraw-Hill Book Company, 1966).CrossRefGoogle Scholar
[13]Giraud, J.. Cohomologie Non-abelienne (Springer-Verlag, 1971).CrossRefGoogle Scholar
[14]Glenn, P.. Realization of cohomology classes in arbitrary exact categories. J. Pure Appl. Alg. 25 (1982), 33–105.CrossRefGoogle Scholar
[15]Gabriel, P. and Zisman, M.. Calculus of Fractions and Homotopy Theory (Springer-Verlag, 1967).CrossRefGoogle Scholar
[16]Hyland, J. M. E., Robinson, E. P. and Rosolini, G.. The discrete objects in the effective topos. (Preprint, University of Cambridge, 1987.)Google Scholar
[17]Johnstone, P. T.. Topos Theory (Academic Press, 1977).Google Scholar
[18]Johnstone, P. T.. Stone Spaces. Cambridge Studies in Advanced Math. no. 3 (Cambridge University Press, 1982).Google Scholar
[19]Johnstone, P. T.. How general is a generalized space? In Aspects of Topology, London Math. Soc. Lecture Notes no. 93 (Cambridge University Press, 1985), pp. 77–111.CrossRefGoogle Scholar
[20]Joyal, A. and Tierney, M.. An Extension of the Galois Theory of Grothendieck. Memoirs Amer. Math. Soc. no. 51 (American Mathematical Society, 1984).CrossRefGoogle Scholar
[21]Moerdijk, I.. Continuous fibrations and inverse limits of toposes. Compositio Math. 58 (1986), 45–72.Google Scholar
[22]Moerdijk, I.. The classifying topos of a continuous groupoid I, II. (Preprints, University of Amsterdam, 1986/1987.)Google Scholar
[23]Moerdijk, I.. Morita equivalence for continuous groups. Math. Proc. Cambridge Philos. Soc. 103 (1988), 97–115.CrossRefGoogle Scholar
[24]Moerdijk, I.. Toposes and groupoids. (Preprint, University of Chicago, 1988.)CrossRefGoogle Scholar
[25]Paré, R.. Indexed Categories and generated topologies. J. Pure Appl. Alg. 19 (1980), 305–400.CrossRefGoogle Scholar
[26]Paré, R. and Schumacher, D.. Abstract families and the adjoint functor theorem. In Indexed Categories and their Applications, Lecture Notes in Math. vol. 661 (Springer-Verlag, 1978), pp. 1–125.CrossRefGoogle Scholar
[27]Pitts, A.. On product and change of base for toposes. (Preprint, University of Sussex.)Google Scholar
[28]Steenrod, N.. The Topology of Fibre Bundles (Princeton University Press, 1951).CrossRefGoogle Scholar