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Published May 16, 2017 | Submitted
Journal Article Open

Polish groupoids and functorial complexity

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.

Additional Information

© 2017 American Mathematical Society. Received by editor(s): March 17, 2016; Received by editor(s) in revised form: September 27, 2016; Published electronically: May 16, 2017. 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. Looking at Polish groupoids was first suggested to the author by George Elliott on the occasion of a joint work with Samuel Coskey and Ilijas Farah. The author would like to thank all of them, as well as Marcin Sabok, Kostyantyn Slutskyy, Anush Tserunyan, for several helpful conversations. The author is also in debt to the anonymous referee for a large number of helpful remarks, which significantly improved the exposition of the present paper.

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