Polish groupoids and functorial complexity

Abstract: We introduce and study the notion of functorial Borel complexity for Polish groupoids. Such a notion aims at measuring 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 coincide with the usual Borel complexity of equivalence relations. Our main result is that on one hand for Polish groupoids with 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.