| Title:
|
A duality between infinitary varieties and algebraic theories (English) |
| Author:
|
Adámek, Jiří |
| Author:
|
Koubek, Václav |
| Author:
|
Velebil, Jiří |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
41 |
| Issue:
|
3 |
| Year:
|
2000 |
| Pages:
|
529-541 |
| . |
| Category:
|
math |
| . |
| Summary:
|
A duality between $\lambda$-ary varieties and $\lambda$-ary algebraic theories is proved as a direct generalization of the finitary case studied by the first author, F.W. Lawvere and J. Rosick'y. We also prove that for every uncountable cardinal $\lambda $, whenever $\lambda $-small products commute with $\Cal D$-colimits in $\text{Set}$, then $\Cal D$ must be a $\lambda $-filtered category. We nevertheless introduce the concept of $\lambda$-sifted colimits so that morphisms between $\lambda$-ary varieties (defined to be $\lambda$-ary, regular right adjoints) are precisely the functors preserving limits and $\lambda$-sifted colimits. (English) |
| Keyword:
|
variety |
| Keyword:
|
Lawvere theory |
| Keyword:
|
sifted colimit |
| Keyword:
|
filtered colimit |
| MSC:
|
08B99 |
| MSC:
|
18A30 |
| MSC:
|
18C10 |
| idZBL:
|
Zbl 1035.08004 |
| idMR:
|
MR1795083 |
| . |
| Date available:
|
2009-01-08T19:04:51Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119187 |
| . |
| Reference:
|
[ALR] Adámek J., Lawvere F.W., Rosický J.: On the duality between varieties and algebraic theories.submitted. |
| Reference:
|
[AP] Adámek J., Porst H.-E.: Algebraic theories of quasivarieties.J. Algebra 208 (1998), 379-398. MR 1655458 |
| Reference:
|
[AR] Adámek J., Rosický J.: Locally Presentable and Accessible Categories.Cambridge University Press, 1994. MR 1294136 |
| Reference:
|
[Bo] Borceux F.: Handbook of Categorical Algebra.Cambridge University Press, 1994, (in three volumes). Zbl 1143.18003 |
| Reference:
|
[GU] Gabriel P., Ulmer F.: Lokal präsentierbare Kategorien.LNM 221, Springer-Verlag, Berlin, 1971. Zbl 0225.18004, MR 0327863 |
| Reference:
|
[L] Lawvere F.W.: Functorial semantics of algebraic theories.Dissertation, Columbia University, 1963. Zbl 1062.18004, MR 0158921 |
| Reference:
|
[S] Street R.: Fibrations in bicategories.Cahiers Topol. Géom. Différentielles Catégoriques XXI (1980), 111-160. Zbl 0436.18005, MR 0574662 |
| . |
