Hostname: page-component-54dcc4c588-2ptsp Total loading time: 0 Render date: 2025-09-16T05:44:39.388Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 ?

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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