Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-27T12:35:24.187Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasicomponents in topos theory: the hyperpure, complete spread factorization

Published online by Cambridge University Press:  12 February 2007

MARTA BUNGE  and
JONATHON FUNK
MARTA BUNGE
Affiliation:
Department of Mathematics, McGill University, Burnside Hall 805 Sherbrooke Street West Montreal, Quebec, CanadaH3A 2K6. e-mail: [email protected]
JONATHON FUNK
Affiliation:
Dept. of Comp. Sci., Maths, and Physics, The University of the West Indies Cave Hill Campus, P.O. Box 64 Bridgetown, Barbados. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish the existence and uniqueness of a factorization for geometric morphisms that generalizes the pure, complete spread factorization for geometric morphisms with a locally connected domain. A complete spread with locally connected domain over a topos is a geometric counterpart of a Lawvere distribution on the topos, and the factorization itself is of the comprehensive type. The new factorization removes the topologically restrictive local connectedness requirement by working with quasicomponents in topos theory. In the special case when the codomain topos of the geometric morphism coincides with the base topos, the factorization gives the locale of quasicomponents of the domain topos, or its ‘0-dimensional’ reflection.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 142 , Issue 1 , January 2007 , pp. 47 - 62
Copyright
Copyright © Cambridge Philosophical Society 2007

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.)

References

REFERENCES

[1] Barr, M. and Paré, R.. Molecular toposes. J. Pure Appl. Alg. 17 (1980), 127–152.CrossRefGoogle Scholar
[2] Bunge, M. and Funk, J.. Spreads and the symmetric topos. J. Pure Appl. Alg. 113 (1996), 1–38.CrossRefGoogle Scholar
[3] Bunge, M. and Funk, J.. Spreads and the symmetric topos II. J. Pure Appl. Alg. 130 (1) (1998), 49–84.CrossRefGoogle Scholar
[4] Bunge, M. and Funk, J.. Singular coverings of toposes. Lecture Notes in Mathematics, vol. 1890 Springer-Verlag, 2006.Google Scholar
[5] Bunge, M., Funk, J., Jibladze, M. and Streicher, T.. The Michael completion of a topos spread. J. Pure Appl. Alg. 175 (2002), 63–91.CrossRefGoogle Scholar
[6] Bunge, M., Funk, J., Jibladze, M. and Streicher, T.. Definable completeness. Cahiers de Top. et Géom. Diff. Catégoriques XLV–4 (2004), 1–24.Google Scholar
[7] Fox, R. H.. Covering spaces with singularities. In Fox, R. H. et al. ., editors, Algebraic Geometry and Topology: A Symposium in Honor of S. Lefschetz (Princeton University Press, 1957) pages 243–257.CrossRefGoogle Scholar
[8] Funk, J.. The locally connected coclosure of a Grothendieck topos. J. Pure Appl. Alg. 137 (1999), 17–27.CrossRefGoogle Scholar
[9] Johnstone, P. T.. Factorization theorems for geometric morphisms II. In Categorical Aspects of Topology and Analysis, Proceedings Carleton University (Ottawa 1980). Lecture Notes in Math. 915 (1982), 216–233.CrossRefGoogle Scholar
[10] Michael, E.. Completing a spread (in the sense of Fox) without local connectedness. Indag. Math. 25 (1963), 629–633.CrossRefGoogle Scholar
[11] Willard, S.. General Topology. Addison-Wesley Series in Mathematics. (Addison-Wesley Publishing Company, 1968).Google Scholar