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Tbilisi Mathematical Journal

Editor-in-Chief: Inassaridze, Hvedri

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Mathematical Citation Quotient (MCQ) 2016: 0.14

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1512-0139
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Pitts monads and a lax descent theorem

Marta Bunge
  • Corresponding author
  • Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 Sherbrooke Street West, Montreal, QC, Canada H3A 2K6
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Published Online: 2015-02-02 | DOI: https://doi.org/10.1515/tmj-2015-0001

Abstract

A theorem of A.M.Pitts (1986) states that essential surjections of toposes bounded over a base topos S are of effective lax descent. The symmetric monad M on the 2-category of toposes bounded over S is a KZ-monad (Bunge-Carboni 1995) and the M-maps are precisely the S-essential geometric morphisms (Bunge-Funk 2006). These last two results led me to conjecture and then prove the general lax descent theorem that is the subject matter of this paper.By a 'Pitts KZ-monad' on a 2-category K it is meant here a locally fully faithful equivariant KZ-monad M on K that is required to satisfy an analogue of Pitts' theorem on bicomma squares along essential geometric morphisms. The main result of this paper states that, for a Pitts KZ-monad M on a 2-category K ('of spaces'), every surjective M-map is of effective lax descent. There is a dual version of this theorem for a Pitts co-KZ-monad N. These theorems have (known and new) consequences regarding (lax) descent for morphisms of toposes and locales.

Keywords: lax descent; Kock-Zoeberlein monads; symmetric monad; coherent toposes; powerlocales; Pitts' theorem.

References

  • [1] D. Ballard and W. Boshuck, Definability and Descent, Journal of Symbolic Logic 63, 2 (1998) 372-378. Google Scholar

  • [2] M. Barr and R. Pare, Molecular toposes, J. Pure and Apl. Algebra 17 (1980) 127-152. Google Scholar

  • [3] J. Benabou and J. Roubaud, Monades et descente, C.R.A.S. 270 (1970) 96-98. Google Scholar

  • [4] M. Bunge and A. Carboni, The symmetric topos, J. Pure and Appl. Alg. 105 (1995) 233-249. Google Scholar

  • [5] M. Bunge and J. Funk, Singular Coverings of Toposes, Lecture Notes in Mathematics 1890, Springer, 2006. Google Scholar

  • [6] M. H. Escardo, Properly injective spaces and function spaces, Topology and Its Applications 89, 1-2 (1998) 75-120. Web of ScienceGoogle Scholar

  • [7] P. T. Johnstone, Topos Theory, London Mathematical Society Monographs 10, Academic Press, 1977. Google Scholar

  • [8] P. T. Johnstone, Sketches on an Elephant. A Topos Theory Compendium,Volumes 1 and 2 Oxford Logic Guides 43, Oxford University Press, 2002. Google Scholar

  • [9] A. Joyal and M. Tierney, An extension of the Galois theory of Grothendieck, Memoirs of the American Mathematical Society 309, 1984. Google Scholar

  • [10] A. Kock, Monads for which structure is adjoint to units, J. Pure and Appl. Alg. 104 (1995) 41-59. Google Scholar

  • [11] M. Korostenski and C. C. A. Labuschagne, Lax proper maps of locales, J. Pure and Appl. Alg. 208 (2007) 655- 664. Web of ScienceGoogle Scholar

  • [12] F. W. Lawvere, Integration on presheaf toposes, Proceedings of the Oberwolfach Meeting on Category Theory, 1966. Google Scholar

  • [13] S. Mac Lane, Categories for the Working Mathematician, Second Edition, Graduate Texts in Mathematics 5, Springer, 1998. Google Scholar

  • [14] I. Moerdijk and J. J. C. Vermeulen, Proof of a conjecture of A. Pitts, J. Pure Appl. Algebra 143 (1999) 329-338. Google Scholar

  • [15] I. Moerdijk and J. J. C. Vermeulen, Proper Maps of Toposes, Memoirs of the American Mathematical Society 148, 705, 2000. Google Scholar

  • [16] A. M. Pitts, On product and change of base for toposes, Cahiers de Top. et Geom. Diff. Categoriques 26-1 (1985) 43-61. Google Scholar

  • [17] A. M. Pitts, Lax descent for essential surjections, Slides for a talk at the Category Theory Conference, Cambridge, July 21, 1986. Google Scholar

  • [18] J. J. C. Vermeulen, Proper maps of locales, J. Pure Appl. Alg, 92 (1994) 79-107. Google Scholar

  • [19] S. J. Vickers. Locales are not pointless, In C. Hankin, I. Mackie, and R. Nagarajan, editors, Theory and Formal Methods 1994. IC Press, 1995. Google Scholar

  • [20] S. J. Vickers, Cosheaves and connectedness in formal topology, Annals of Pure and Applied Logic 63 (2012) 157-174. Web of ScienceGoogle Scholar

  • [21] M. W. Zawadowski, Descent and duality, Ann. Pure Appl. Logic 71 (1995) 131-188. Google Scholar

About the article

Published Online: 2015-02-02


Citation Information: Tbilisi Mathematical Journal, Volume 8, Issue 1, ISSN (Online) 1512-0139, DOI: https://doi.org/10.1515/tmj-2015-0001.

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© 2015 Marta Bunge. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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