Initial normal covers in bi-Heyting toposes

Francis Borceux; Dominique Bourn; Peter Johnstone

Archivum Mathematicum (2006)

  • Volume: 042, Issue: 4, page 335-356
  • ISSN: 0044-8753

Abstract

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The dual of the category of pointed objects of a topos is semi-abelian, thus is provided with a notion of semi-direct product and a corresponding notion of action. In this paper, we study various conditions for representability of these actions. First, we show this to be equivalent to the existence of initial normal covers in the category of pointed objects of the topos. For Grothendieck toposes, actions are representable provided the topos admits an essential Boolean covering. This contains the case of Boolean toposes and toposes of presheaves. In the localic case, the representability of actions forces the topos to be bi-Heyting: the lattices of subobjects are both Heyting algebras and the dual of Heyting algebras.

How to cite

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Borceux, Francis, Bourn, Dominique, and Johnstone, Peter. "Initial normal covers in bi-Heyting toposes." Archivum Mathematicum 042.4 (2006): 335-356. <http://eudml.org/doc/249831>.

@article{Borceux2006,
abstract = {The dual of the category of pointed objects of a topos is semi-abelian, thus is provided with a notion of semi-direct product and a corresponding notion of action. In this paper, we study various conditions for representability of these actions. First, we show this to be equivalent to the existence of initial normal covers in the category of pointed objects of the topos. For Grothendieck toposes, actions are representable provided the topos admits an essential Boolean covering. This contains the case of Boolean toposes and toposes of presheaves. In the localic case, the representability of actions forces the topos to be bi-Heyting: the lattices of subobjects are both Heyting algebras and the dual of Heyting algebras.},
author = {Borceux, Francis, Bourn, Dominique, Johnstone, Peter},
journal = {Archivum Mathematicum},
keywords = {semi-abelian category},
language = {eng},
number = {4},
pages = {335-356},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Initial normal covers in bi-Heyting toposes},
url = {http://eudml.org/doc/249831},
volume = {042},
year = {2006},
}

TY - JOUR
AU - Borceux, Francis
AU - Bourn, Dominique
AU - Johnstone, Peter
TI - Initial normal covers in bi-Heyting toposes
JO - Archivum Mathematicum
PY - 2006
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 042
IS - 4
SP - 335
EP - 356
AB - The dual of the category of pointed objects of a topos is semi-abelian, thus is provided with a notion of semi-direct product and a corresponding notion of action. In this paper, we study various conditions for representability of these actions. First, we show this to be equivalent to the existence of initial normal covers in the category of pointed objects of the topos. For Grothendieck toposes, actions are representable provided the topos admits an essential Boolean covering. This contains the case of Boolean toposes and toposes of presheaves. In the localic case, the representability of actions forces the topos to be bi-Heyting: the lattices of subobjects are both Heyting algebras and the dual of Heyting algebras.
LA - eng
KW - semi-abelian category
UR - http://eudml.org/doc/249831
ER -

References

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  12. Janelidze G., Márki L., Tholen W., Semi-abelian categories, J. Pure Appl. Alg. 168 (2002), 367–386. Zbl0993.18008MR1887164
  13. Johnstone P. T., Stone Spaces, Cambridge Stud. Adv. Math. No. 3 (1982). (1982) Zbl0499.54001MR0698074
  14. Johnstone P. T., Sketches of an Elephant: a Topos Theory Compendium, volumes 1–2, Oxford Logic Guides 43–44 (2002). Zbl1071.18002MR1953060
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