Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T22:02:25.616Z Has data issue: false hasContentIssue false
  • English
  • Français

CLASSIFICATION THEORY FOR ACCESSIBLE CATEGORIES

Published online by Cambridge University Press:  09 March 2016

M. LIEBERMAN  and
J. ROSICKÝ
M. LIEBERMAN
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS MASARYK UNIVERSITY FACULTY OF SCIENCES KOTLÁŘSKÁ 2, 611 37 BRNO CZECH REPUBLICE-mail: [email protected]
J. ROSICKÝ
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS MASARYK UNIVERSITY FACULTY OF SCIENCES KOTLÁŘSKÁ 2, 611 37 BRNO CZECH REPUBLICE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that a number of results on abstract elementary classes (AECs) hold in accessible categories with concrete directed colimits. In particular, we prove a generalization of a recent result of Boney on tameness under a large cardinal assumption. We also show that such categories support a robust version of the Ehrenfeucht–Mostowski construction. This analysis has the added benefit of producing a purely language-free characterization of AECs, and highlights the precise role played by the coherence axiom.

Keywords

model theorycategorical logicabstract elementary classaccessible categoryGalois typestameness
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 1 , March 2016 , pp. 151 - 165
Copyright
Copyright © The Association for Symbolic Logic 2016 

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

Adámek, J. and Rosický, J., Locally Presentable and Accessible Categories, Cambridge University Press, Cambridge, 1994.Google Scholar
Baldwin, J., Categoricity, AMS, Providence, Rhode Island, 2009.Google Scholar
Baldwin, J., Ehrenfeucht-Mostowski models in abstract elementary classes, Logic and Its Applications (Zhang, Y. and Blass, A., editors), Contemporary Mathematics, vol. 380, AMS, Providence, Rhode Island, 2005.CrossRefGoogle Scholar
Baldwin, J. and Shelah, S., Examples of non-locality, this Journal, vol. 73 (2008), pp. 765–782.Google Scholar
Beke, T. and Rosický, J., Abstract elementary classes and accessible categories. Annals of Pure and Applied Logic, vol. 163 (2012), pp. 2008–2017.CrossRefGoogle Scholar
Boney, W., Tameness from large cardinal axioms. this Journal, vol. 79 (2014), no. 4, pp. 1092–1119.Google Scholar
Grossberg, R., Classification theory for abstract elementary classes, Logic and Algebra (Zhang, Yi, editor), vol. 302, AMS, Providence, Rhode Island 2002.CrossRefGoogle Scholar
Grossberg, R. and VanDieren, M., Categoricity from one successor cardinal in tame abstract elementary classes. Journal of Mathematical Logic, vol. 143 (2006), pp. 103–138.Google Scholar
Hyttinen, T. and Kesälä, M, Independence in finitary abstract elementary classes. Annals of Pure and Applied Logic, vol. 143 (2006), pp. 103–138.Google Scholar
Jech, T., Set Theory, Academic Press, New York, 1978.Google Scholar
Lieberman, M., Category theoretic aspects of abstract elementary classes. Annals of Pure and Applied Logic, vol. 162 (2011), pp. 903–915.Google Scholar
Kueker, D. W., Abstract elementary classes and infinitary logic. Annals of Pure and Applied Logic, vol. 156 (2008), pp. 274–286.Google Scholar
Makkai, M. and Paré, R., Accessible Categories: The Foundations of Categorical Model Theory, AMS, Providence, Rhode Island, 1989.Google Scholar
Rosický, J., Concrete categories and infinitary languages. Journal of Pure and Applied Algebra, vol. 22 (1981), pp. 309–339.CrossRefGoogle Scholar
Rosický, J., Accessible categories, saturation and categoricity, this Journal, vol. 62 (1997), pp. 891–901.Google Scholar
Rosický, J., On combinatorial model categories. Applied Categorical Structures, vol. 17 (2009), pp. 303–316.Google Scholar
Shelah, S., Classification of nonelementary classes II, abstract elementary classes, Lecture Notes in Mathematics, vol. 1292 (1987), pp. 419–497, Springer-Verlag, Berlin, 1987.Google Scholar
Shelah, S., Categoricity of abstract elementary classes: going up inductive step, preprint. arXiv:0011215v2.Google Scholar