Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T18:53:35.698Z Has data issue: false hasContentIssue false
  • English
  • Français

On Colimits and Elementary Embeddings

Published online by Cambridge University Press:  12 March 2014

Joan Bagaria  and
Andrew Brooke-Taylor
Joan Bagaria
Affiliation:
Icrea (Institució Catalana de Recerca i Estudis Avancats), Departament de Lògica, Història i Filosofia de la Ciència, Universitat de Barcelona, Montalegre 6, 08001 Barcelona, Catalonia, Spain, E-mail:[email protected]
Andrew Brooke-Taylor
Affiliation:
Group of Logic, Statistics & Informatics, Graduate School of System Informatics, Kobe University, Rokko-Dai 1-1, Nada, Kobe, 657-0013, Japan, E-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a sharper version of a theorem of Rosický, Trnková and Adámek [13], and a new proof of a theorem of Rosický [12], both about colimits in categories of structures. Unlike the original proofs, which use category-theoretic methods, we use set-theoretic arguments involving elementary embeddings given by large cardinals such as α-strongly compact and C(n)-extendible cardinals.

Keywords

Colimitselementary embeddingsVopěnka's Principlestrongly compact cardinalsaccessible categories
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 2 , June 2013 , pp. 562 - 578
Copyright
Copyright © Association for Symbolic Logic 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Adámek, Jiří and Rosický, Jiří, Locally presentable and accessible categories, London Mathematical Society Lecture Note Series, no. 189, Cambridge University Press, Cambridge, 1994.CrossRefGoogle Scholar
[2] Bagaria, Joan, C(n)-cardinals. Archive for Mathematical Logic, vol. 51 (2012), pp. 213–240.CrossRefGoogle Scholar
[3] Bagaria, Joan, Casacuberta, Carles, Mathias, Adrian R. D., and Rosický, Jiří, Definable orthogonality classes in accessible categories are small, Journal of the European Mathematical Society, to appear.Google Scholar
[4] Bagaria, Joan and Magidor, Menachem, Group radicals and strongly compact cardinals, Transactions of the American Mathematical Society, to appear.Google Scholar
[5] Casacuberta, Carles, Scevenels, Dirk, and Smith, Jeffrey H., Implications of large-cardinal principles in homotopical localization, Advances in Mathematics, vol. 197 (2005), pp. 120–139.CrossRefGoogle Scholar
[6] Eda, Katsuya and Abe, Yoshihiro, Compact cardinals and abelian groups, Tsukuba Journal of Mathematics, vol. 11 (1987), no. 2, pp. 353–360.Google Scholar
[7] Kanamori, Akihiro, The higher infinite, 2nd ed.. Springer, 2003.Google Scholar
[8] Kunen, Kenneth, Elementary embeddings and infinitary combinatorics, this Journal, vol. 36 (1971), pp. 407–413.Google Scholar
[9] Lane, Saunders Mac, Categories for the working mathematician, second ed., Graduate Texts in Mathematics, vol. 5, Springer, New York, 1978.CrossRefGoogle Scholar
[10] Pultr, A. and Trnková, V., Combinatorial, algebraic and topological representations of groups, semigroups and categories, North-Holland, Amsterdam, 1980.Google Scholar
[11] Richter, Michael, Limites in Kategorien von Relationalsystemen, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 17 (1971), pp. 75–90.CrossRefGoogle Scholar
[12] Rosický, J., More on directed colimits of models. Applied Categorical Structures, vol. 2 (1994), pp. 71–76.CrossRefGoogle Scholar
[13] Rosický, J., Trnková, V., and Adámek, J., Unexpected properties of locally presentable categories, Algebra Universalis, vol. 27 (1990), no. 2, pp. 153–170.CrossRefGoogle Scholar