Componentwise Polish groupoids and equivalence relations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908436673855488 |
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| author | Chen, Ruiyuan |
| author_facet | Chen, Ruiyuan |
| contents | We study Borel equivalence relations equipped with a uniformly Borel family of Polish topologies on each equivalence class, and more generally, standard Borel groupoids equipped with such a family of topologies on each connected component. Such "componentwise Polish topologies" capture precisely the topological information determined by the Borel structure of a Polish group action, by the Becker--Kechris theorem. We prove that conversely, every abstract such Borel componentwise Polish groupoid obeying suitable axioms admits a Borel equivalence of groupoids to a global open Polish groupoid. Together with known results, this implies that every such groupoid is Borel equivalent to an action groupoid of a Polish group action; in particular, the induced equivalence relations are Borel bireducible.
Our results are also valid for Borel groupoids with componentwise quasi-Polish topologies; and under stronger uniformity assumptions, we show that such groupoids in fact themselves admit global quasi-Polish topologies. As a byproduct, we also generalize several standard tools for Polish groups and their actions to the setting of componentwise quasi-Polish groupoids, including Vaught transforms, Effros's theorem on orbits, and the open mapping theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04138 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Componentwise Polish groupoids and equivalence relations Chen, Ruiyuan Logic Dynamical Systems 03E15, 22A22, 22F10 We study Borel equivalence relations equipped with a uniformly Borel family of Polish topologies on each equivalence class, and more generally, standard Borel groupoids equipped with such a family of topologies on each connected component. Such "componentwise Polish topologies" capture precisely the topological information determined by the Borel structure of a Polish group action, by the Becker--Kechris theorem. We prove that conversely, every abstract such Borel componentwise Polish groupoid obeying suitable axioms admits a Borel equivalence of groupoids to a global open Polish groupoid. Together with known results, this implies that every such groupoid is Borel equivalent to an action groupoid of a Polish group action; in particular, the induced equivalence relations are Borel bireducible. Our results are also valid for Borel groupoids with componentwise quasi-Polish topologies; and under stronger uniformity assumptions, we show that such groupoids in fact themselves admit global quasi-Polish topologies. As a byproduct, we also generalize several standard tools for Polish groups and their actions to the setting of componentwise quasi-Polish groupoids, including Vaught transforms, Effros's theorem on orbits, and the open mapping theorem. |
| title | Componentwise Polish groupoids and equivalence relations |
| topic | Logic Dynamical Systems 03E15, 22A22, 22F10 |
| url | https://arxiv.org/abs/2507.04138 |