| Title:
|
On the injectivity of Boolean algebras (English) |
| Author:
|
Banaschewski, B. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
34 |
| Issue:
|
3 |
| Year:
|
1993 |
| Pages:
|
501-511 |
| . |
| Category:
|
math |
| . |
| Summary:
|
The functor taking global elements of Boolean algebras in the topos $\text{$\bold{Sh}\frak B$}$ of sheaves on a complete Boolean algebra $\frak B$ is shown to preserve and reflect injectivity as well as completeness. This is then used to derive a result of Bell on the Boolean Ultrafilter Theorem in $\frak B$-valued set theory and to prove that (i) the category of complete Boolean algebras and complete homomorphisms has no non-trivial injectives, and (ii) the category of frames has no absolute retracts. (English) |
| Keyword:
|
sheaves on a complete Boolean algebra |
| Keyword:
|
injective Boolean algebra |
| Keyword:
|
complete Boolean algebra |
| Keyword:
|
injective complete Boolean algebra |
| Keyword:
|
absolute frame retract |
| MSC:
|
03E25 |
| MSC:
|
03E40 |
| MSC:
|
03G05 |
| MSC:
|
06A23 |
| MSC:
|
06E10 |
| MSC:
|
06E99 |
| MSC:
|
18B25 |
| MSC:
|
18B99 |
| idZBL:
|
Zbl 0789.06007 |
| idMR:
|
MR1243081 |
| . |
| Date available:
|
2009-01-08T18:05:38Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118606 |
| . |
| Reference:
|
[1] Banaschewski B.: On pushing out frames.Comment. Math. Univ. Carolinae 31 (1990), 13-21. Zbl 0706.18003, MR 1056165 |
| Reference:
|
[2] Banaschewski B., Bhutani K.R.: Boolean algebras in a localic topos.Math. Proc. Cambridge Phil. Soc. 100 (1986), 43-55. Zbl 0598.18001, MR 0838652 |
| Reference:
|
[3] Bell J.L.: On the strength of the Sikorski Extension Theorem for Boolean algebras.J. Symb. Logic 48 (1983), 841-846. Zbl 0537.03032, MR 0716646 |
| Reference:
|
[4] Blass A., Sčedrov A.: Freyd's models for the independence of the axiom of choice.Mem. Amer. Math. Soc. 79 (1989), No. 404. Zbl 0687.03031 |
| Reference:
|
[5] Higgs D.: A category approach to Boolean-valued set theory.preprint, University of Waterloo, 1973. |
| Reference:
|
[6] Johnstone P.T.: Topos theory.L.M.S. Mathematical Monographs no. 10, Academic Press, 1977. Zbl 1071.18002, MR 0470019 |
| Reference:
|
[7] Johnstone P.T.: Conditions related to De Morgan's law.Springer LNM 253 (1979), 479-491. Zbl 0445.03041, MR 0555556 |
| Reference:
|
[8] Johnstone P.T.: Stone spaces.Cambridge Studies in Advanced Mathematics 3, Cambridge University Press, Cambridge, 1982. Zbl 0586.54001, MR 0698074 |
| Reference:
|
[9] Mac Lane S.: Categories for the Working Mathematician.Graduate Texts in Mathematics 5, Springer-Verlag, Berlin, Heidelberg, New York, 1971. Zbl 0906.18001, MR 0354798 |
| Reference:
|
[10] Pultr A.: Oral communication, October 1986.. |
| . |
