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CATEGORICAL SEMANTICS OF METRIC SPACES AND CONTINUOUS LOGIC

Part of: Algebraic logic Spaces with richer structures Categories and geometry General logic Special categories

Published online by Cambridge University Press:  05 October 2020

SIMON CHO
SIMON CHO*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF MICHIGAN 530 CHURCH STREET ANN ARBOR, MI48109, USAE-mail: cho.simon.math@gmail.comE-mail: simoncho@umich.edu
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Abstract

Using the category of metric spaces as a template, we develop a metric analogue of the categorical semantics of classical/intuitionistic logic, and show that the natural notion of predicate in this “continuous semantics” is equivalent to the a priori separate notion of predicate in continuous logic, a logic which is independently well-studied by model theorists and which finds various applications. We show this equivalence by exhibiting the real interval $[0,1]$ in the category of metric spaces as a “continuous subobject classifier” giving a correspondence not only between the two notions of predicate, but also between the natural notion of quantification in the continuous semantics and the existing notion of quantification in continuous logic.

Along the way, we formulate what it means for a given category to behave like the category of metric spaces, and afterwards show that any such category supports the aforementioned continuous semantics. As an application, we show that categories of presheaves of metric spaces are examples of such, and in fact even possess continuous subobject classifiers.

Keywords

categorical semanticscontinuous logicmetric spacessubobject classifierpresheaves of metric spacesquantales

MSC classification

Primary: 03G30: Categorical logic, topoi
Secondary: 03B50: Many-valued logic 18B30: Categories of topological spaces and continuous mappings 18F20: Presheaves and sheaves 54E35: Metric spaces, metrizability
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 3 , September 2020 , pp. 1044 - 1078
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Copyright
© The Association for Symbolic Logic 2020

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References

REFERENCES

Avigad, J. and Iovino, J., Ultraproducts and metastability . New York Journal of Mathematics , vol. 19 (2013), pp. 713–727.Google Scholar
Čech, E., Topological Spaces . Revised edition by Frolík, Z. and Katětov, M.. Scientific editor, Pták, V.. Editor of the English translation, Junge, C. O.. Publishing House of the Czechoslovak Academy of Sciences, Prague; Interscience Publishers John Wiley & Sons, London-New York-Sydney, 1966.Google Scholar
Cho, S., An application of continuous logic to fixed point theory, preprint, 2019, arXiv:1610.05397.Google Scholar
Farah, I., Hart, B., and Sherman, D., Model theory of operator algebras II: Model theory . Israel Journal of Mathematics , vol. 201 (2014), no. 1, pp. 477–505.CrossRefGoogle Scholar
Farah, I., Hart, B., and Sherman, D., Model theory of operator algebras III: Elementary equivalence and $I{I}_1$ factors . Bulletin of the London Mathematical Society , vol. 46 (2014), no. 3, pp. 609–628.CrossRefGoogle Scholar
Freyd, P. J. and Scedrov, A., Categories, Allegories , North-Holland Mathematical Library, vol. 39, North-Holland, Amsterdam, 1990.Google Scholar
Johnstone, P. T., Sketches of an Elephant: A Topos Theory Compendium, vol. 2 , Oxford Logic Guides, vol. 44, Clarendon Press, Oxford University Press, Oxford, 2002.Google Scholar
Lane, S. M. and Moerdijk, I., Sheaves in Geometry and Logic: A First Introduction to Topos Theory , Universitext, Springer-Verlag, New York, NY, 1994. Corrected reprint of the 1992 edition.CrossRefGoogle Scholar
Lawvere, F. W., Metric spaces, generalized logic, and closed categories . Rendiconti del seminario matématico e fisico di Milano , vol. 43 (1973), no. 1, pp. 135–166 (1974); Repr. Theory Appl. Categ., (1):1–37, 2002. With an author commentary: Enriched categories in the logic of geometry and analysis.CrossRefGoogle Scholar
Makkai, M. and Reyes, G. E., First Order Categorical Logic: Model-Theoretical Methods in the Theory of Topoi and Related Categories , Lecture Notes in Mathematics, vol. 611, Springer-Verlag, Berlin, 1977.CrossRefGoogle Scholar
Robinson, M., Sheaves are the canonical data structure for sensor integration . Information Fusion , vol. 36 (2017), pp. 208–224.CrossRefGoogle Scholar
Robinson, M., Assignments to sheaves of pseudometric spaces, preprint, 2018, arXiv:1805.08927.Google Scholar
Yaacov, I. B., Stability and stable groups in continuous logic , this Journal, vol. 75 (2010), no. 3, pp. 1111–1136.Google Scholar
Yaacov, I. B. and Usvyatsov, A., Continuous first order logic and local stability . Transactions of the American Mathematical Society , vol. 362 (2010), no. 10, pp. 5213–5259.CrossRefGoogle Scholar
Yaacov, I. B., Berenstein, A., Henson, C. W., and Usvyatsov, A., Model theory for metric structures . Model Theory with Applications to Algebra and Analysis , vol. 2 (2008), pp. 315–427.CrossRefGoogle Scholar