| Title:
|
Initial normal covers in bi-Heyting toposes (English) |
| Author:
|
Borceux, Francis |
| Author:
|
Bourn, Dominique |
| Author:
|
Johnstone, Peter |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
42 |
| Issue:
|
4 |
| Year:
|
2006 |
| Pages:
|
335-356 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| MSC:
|
06D20 |
| MSC:
|
18B25 |
| idZBL:
|
Zbl 1164.18301 |
| idMR:
|
MR2283017 |
| . |
| Date available:
|
2008-06-06T22:48:45Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/108012 |
| . |
| Reference:
|
[1] Borceux F.: When is $\Omega $ a cogenerator in a topos?.Cahiers Topol. Géom. Diff. 16 (1975), 3–15. Zbl 0311.18006, MR 0382393 |
| Reference:
|
[2] Borceux F.: Handbook of Categorical Algebra 3: Categories of Sheaves.Encyclopaedia Math. Appl. 52 (1994). Zbl 0911.18001, MR 1315049 |
| Reference:
|
[3] Borceux F.: A survey of semi-abelian categories.In: Galois theory, Hopf Algebras, and Semi-abelian Categories, Fields Inst. Commun. 43 (2004), 27–60. Zbl 1067.18010, MR 2075580 |
| Reference:
|
[4] Borceux F., Bourn D.: Mal’cev, Protomodular, Homological and Semi-abelian Categories.Math. Appl. 566 (2004). Zbl 1061.18001, MR 2044291 |
| Reference:
|
[5] Borceux F., Bourn D.: Split extension classifier and centrality.to appear in the Proceedings of the Streetfest 2005. Zbl 1133.18002, MR 2342823 |
| Reference:
|
[6] Borceux F., Janelidze G., Kelly G. M.: Internal object actions.Comment. Math. Univ. Carolin. 46 (2005), 235–255. Zbl 1121.18004, MR 2176890 |
| Reference:
|
[7] Borceux F., Janelidze G., Kelly G. M.: On the representability of actions in a semi-abelian category.Theory Appl. Categ. 14 (2005), 244–286. Zbl 1103.18006, MR 2182676 |
| Reference:
|
[8] Bourn D.: Normal functors and strong protomodularity.Theory Appl. Categ. 7 (2000), 206–218. Zbl 0947.18004, MR 1766393 |
| Reference:
|
[9] Bourn D.: A categorical genealogy for the congruence distributive property.Theory Appl. Categ. 8 (2001), 391–407. Zbl 0978.18005, MR 1847038 |
| Reference:
|
[10] Bourn D.: Protomodular aspects of the dual of a topos.Adv. Math. 187 (2004), 240–255. MR 2074178 |
| Reference:
|
[11] Bourn D., Janelidze G.: Protomodularity, descent and semi-direct products.Theory Appl. Categ. 4 (1998), 37–46. MR 1615341 |
| Reference:
|
[12] Janelidze G., Márki L., Tholen W.: Semi-abelian categories.J. Pure Appl. Alg. 168 (2002), 367–386. Zbl 0993.18008, MR 1887164 |
| Reference:
|
[13] Johnstone P. T.: Stone Spaces.Cambridge Stud. Adv. Math. No. 3 (1982). Zbl 0499.54001, MR 0698074 |
| Reference:
|
[14] Johnstone P. T.: Sketches of an Elephant: a Topos Theory Compendium.volumes 1–2, Oxford Logic Guides 43–44 (2002). Zbl 1071.18002, MR 1953060 |
| Reference:
|
[15] Mac Lane S.: Categories for the Working Mathematician.Graduate Texts in Math. No. 5 (1971; revised edition 1998). Zbl 0232.18001 |
| . |
