Initial normal covers in bi-Heyting toposes
Borceux, Francis ; Bourn, Dominique ; Johnstone, Peter
Archivum Mathematicum, Tome 042 (2006), p. 335-356 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

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.

Publié le : 2006-01-01
Classification:  06D20,  18B25 
@article{108012,
     author = {Francis Borceux and Dominique Bourn and Peter Johnstone},
     title = {Initial normal covers in bi-Heyting toposes},
     journal = {Archivum Mathematicum},
     volume = {042},
     year = {2006},
     pages = {335-356},
     zbl = {1164.18301},
     mrnumber = {2283017},
     language = {en},
     url = {http://dml.mathdoc.fr/item/108012}
}

Borceux, Francis; Bourn, Dominique; Johnstone, Peter. Initial normal covers in bi-Heyting toposes. Archivum Mathematicum, Tome 042 (2006) pp. 335-356. http://gdmltest.u-ga.fr/item/108012/

Borceux F. When is $\Omega $ a cogenerator in a topos?, Cahiers Topol. Géom. Diff. 16 (1975), 3–15. (1975) | MR 0382393 | Zbl 0311.18006

Borceux F. Handbook of Categorical Algebra 3: Categories of Sheaves, Encyclopaedia Math. Appl. 52 (1994). (1994) | MR 1315049 | Zbl 0911.18001

Borceux F. A survey of semi-abelian categories, In: Galois theory, Hopf Algebras, and Semi-abelian Categories, Fields Inst. Commun. 43 (2004), 27–60. | MR 2075580 | Zbl 1067.18010

Borceux F.; Bourn D. Mal’cev, Protomodular, Homological and Semi-abelian Categories, Math. Appl. 566 (2004). | MR 2044291 | Zbl 1061.18001

Borceux F.; Bourn D. Split extension classifier and centrality, to appear in the Proceedings of the Streetfest 2005. | MR 2342823 | Zbl 1133.18002

Borceux F.; Janelidze G.; Kelly G. M. Internal object actions, Comment. Math. Univ. Carolin. 46 (2005), 235–255. | MR 2176890 | Zbl 1121.18004

Borceux F.; Janelidze G.; Kelly G. M. On the representability of actions in a semi-abelian category, Theory Appl. Categ. 14 (2005), 244–286. | MR 2182676 | Zbl 1103.18006

Bourn D. Normal functors and strong protomodularity, Theory Appl. Categ. 7 (2000), 206–218. | MR 1766393 | Zbl 0947.18004

Bourn D. A categorical genealogy for the congruence distributive property, Theory Appl. Categ. 8 (2001), 391–407. | MR 1847038 | Zbl 0978.18005

Bourn D. Protomodular aspects of the dual of a topos, Adv. Math. 187 (2004), 240–255. | MR 2074178

Bourn D.; Janelidze G. Protomodularity, descent and semi-direct products, Theory Appl. Categ. 4 (1998), 37–46. (1998) | MR 1615341

Janelidze G.; Márki L.; Tholen W. Semi-abelian categories, J. Pure Appl. Alg. 168 (2002), 367–386. | MR 1887164 | Zbl 0993.18008

Johnstone P. T. Stone Spaces, Cambridge Stud. Adv. Math. No. 3 (1982). (1982) | MR 0698074 | Zbl 0499.54001

Johnstone P. T. Sketches of an Elephant: a Topos Theory Compendium, volumes 1–2, Oxford Logic Guides 43–44 (2002). | MR 1953060 | Zbl 1071.18002

Mac Lane S. Categories for the Working Mathematician, Graduate Texts in Math. No. 5 (1971; revised edition 1998). (1971) | Zbl 0232.18001