Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T19:59:40.717Z Has data issue: false hasContentIssue false
  • English
  • Français

A SYNTACTIC CHARACTERIZATION OF MORITA EQUIVALENCE

Published online by Cambridge University Press:  09 January 2018

DIMITRIS TSEMENTZIS
DIMITRIS TSEMENTZIS*
Affiliation:
DEPARTMENT OF PHILOSOPHY PRINCETON UNIVERSITY PRINCETON, NJ08544, USAE-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We characterize Morita equivalence of theories in the sense of Johnstone in terms of a new syntactic notion of a common definitional extension developed by Barrett and Halvorson for cartesian, regular, coherent, geometric and first-order theories. This provides a purely syntactic characterization of the relation between two theories that have equivalent categories of models naturally in any Grothendieck topos.

Keywords

Primary 03G3018B25Secondary 03C52classifying toposesMorita equivalencecommon definitional extension
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 4 , December 2017 , pp. 1181 - 1198
Copyright
Copyright © The Association for Symbolic Logic 2017 

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

Awodey, S. and Bauer, A., Propositions as [types]. Journal of Logic and Computation, vol. 14 (2004), no. 4, pp. 447–471.Google Scholar
Awodey, S. and Forsell, H., First-order logical duality. Annals of Pure and Applied Logic, vol. 164 (2013), no. 3, pp. 319–348.Google Scholar
Barrett, T. and Halvorson, H., Morita equivalence. Review of Symbolic Logic, vol. 9 (2016), no. 3, pp. 556–582.Google Scholar
Butz, C. and Johnstone, P., Classifying toposes for first-order theories. Annals of Pure and Applied Logic, vol. 91 (1998), no. 1, pp. 33–58.Google Scholar
Caramello, O., Syntactic characterizations of properties of classifying toposes. Theory and Applications of Categories, vol. 26 (2012), no. 6, pp. 176–193.Google Scholar
Caramello, O., Universal models and definability. Mathematical Proceedings of the Cambridge Philosophical Society, vol. 152 (2012), no. 2, pp. 279–302.Google Scholar
Caramello, O., Atomic toposes and countable categoricity. Applied Categorical Structures, vol. 20 (2012), no. 4, pp. 379–391.Google Scholar
Forssell, H. and Lumsdaine, P. L., Constructive reflection principles for regular theories, 2016, arXiv preprint arXiv:1604.03851.Google Scholar
Gambino, N. and Aczel, P., The generalized type-theoretic interpretation of constructive set theory, this Journal, vol. 71 (2006), pp. 67–103.Google Scholar
Johnstone, P., Sketches of an Elephant: A Topos Theory Compendium, Oxford University Press, Oxford, 2003.Google Scholar
Maietti, M. E., Modular correspondence between dependent type theories and categories including topoi and pretopoi. Mathematical Structures in Computer Science, vol. 15 (2005), no. 6, pp. 1089–1149.Google Scholar
Maietti, M. E. and Sambin, G., Towards a minimalist foundation for constructive mathematics, From Sets and Types to Topology and Analysis (Crosilla, L. and Schuster, P., editors), Oxford Logic Guides, Clarendon Press, London, 2005, pp. 91–114.CrossRefGoogle Scholar
Makkai, M., Duality and Definability in First-Order Logic, Memoirs of the AMS, vol. 503, American Mathematical Society, Providence, RI, 1993.Google Scholar
Makkai, M., Stone duality for first-order logic. Advances in Mathematics, vol. 65 (1987), no. 2, pp. 97–170.Google Scholar
Makkai, M. and Reyes, G., First-Order Categorical Logic, Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, New York, 1971.Google Scholar
Moerdijk, I., The classifying topos of a continuous groupoid I. Transactions of the American Mathematical Society, vol. 310 (1988), no. 2, pp. 629–668.Google Scholar
Moerdijk, I., The classifying topos of a continuous groupoid II. Cahiers de Topologie et Geometrie Differentielle Categoriques, vol. 31 (1990), no. 2, pp. 137–168.Google Scholar
Moerdijk, I., Toposes and groupoids. Categorical Algebra and its Applications, vol. 1348 (1988), pp. 280–298.Google Scholar
Moerdijk, I. and Mac Lane, S., Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Universitext, Springer, 1994.Google Scholar
Moerdijk, I. and Palmgren, E., Type theories, toposes and constructive set theory: Predicative aspects of AST. Annals of Pure and Applied Logic, vol. 114 (2002), pp. 155–201.Google Scholar
Mycielski, J., A lattice of interpretability types of theories, this Journal, vol. 42 (1977), no. 2, pp. 297–305.Google Scholar
Pinter, C. C., Properties preserved under definitional equivalence and interpretations. Mathematical Logic Quarterly, vol. 24 (1978), pp. 481–488.Google Scholar
Sambin, G. and Valentini, S., Building up a toolbox for Martin-Löf type theory: Subset theory, Twenty-Five Years of Constructive Type Theory (Sambin, G. and Smith, J. M., editors), Clarendon Press, Oxford, 1998, pp. 221–244.Google Scholar