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Polish groupoids and functorial complexity


Author: Martino Lupini; with an appendix by Anush Tserunyan
Journal: Trans. Amer. Math. Soc. 369 (2017), 6683-6723
MSC (2010): Primary 03E15, 22A22; Secondary 54H05
DOI: https://doi.org/10.1090/tran/7102
Published electronically: May 16, 2017
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Abstract: We introduce and study the notion of functorial Borel complexity for Polish groupoids. Such a notion aims to measure the complexity of classifying the objects of a category in a constructive and functorial way. In the particular case of principal groupoids such a notion coincides with the usual Borel complexity of equivalence relations. Our main result is that on one hand for Polish groupoids with an essentially treeable orbit equivalence relation, functorial Borel complexity coincides with the Borel complexity of the associated orbit equivalence relation. On the other hand, for every countable equivalence relation $ E$ that is not treeable there are Polish groupoids with different functorial Borel complexity both having $ E$ as orbit equivalence relation. In order to obtain such a conclusion we generalize some fundamental results about the descriptive set theory of Polish group actions to actions of Polish groupoids, answering a question of Arlan Ramsay. These include the Becker-Kechris results on Polishability of Borel $ G $-spaces, existence of universal Borel $ G$-spaces, and characterization of Borel $ G$-spaces with Borel orbit equivalence relations.


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Additional Information

Martino Lupini
Affiliation: Department of Mathematics and Statistics, N520 Ross, 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada
Address at time of publication: Department of Mathematics, California Institute of Technology, 1200 E. California Boulevard, MC 253-37, Pasadena, California 91125
Email: lupini@caltech.edu

Anush Tserunyan
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 273 Altgeld Hall, 1409 W. Green Street (MC-382), Urbana, Illinois 61801

DOI: https://doi.org/10.1090/tran/7102
Keywords: Polish groupoids, Borel reducibility, Vaught transform
Received by editor(s): March 17, 2016
Received by editor(s) in revised form: September 27, 2016
Published electronically: May 16, 2017
Additional Notes: The author was supported by the York University Elia Scholars Program. This work was completed when the author was attending the Thematic Program on Abstract Harmonic Analysis, Banach and Operator Algebras at the Fields Institute. The hospitality of the Fields Institute is gratefully acknowledged.
Article copyright: © Copyright 2017 American Mathematical Society

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