Structural, point-free, non-Hausdorff topological realization of Borel groupoid actions
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866909137341775872 |
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| author | Chen, Ruiyuan |
| author_facet | Chen, Ruiyuan |
| contents | We extend the Becker--Kechris topological realization and change-of-topology theorems for Polish group actions in several directions. For Polish group actions, we prove a single result that implies the original Becker--Kechris theorems, as well as Sami's and Hjorth's sharpenings adapted levelwise to the Borel hierarchy; automatic continuity of Borel actions via homeomorphisms; and the equivalence of "potentially open" versus "orbitwise open" Borel sets. We also characterize "potentially open" $n$-ary relations, thus yielding a topological realization theorem for invariant Borel first-order structures. We then generalize to groupoid actions, and prove a result subsuming Lupini's Becker--Kechris-type theorems for open Polish groupoids, newly adapted to the Borel hierarchy, as well as topological realizations of actions on fiberwise topological bundles and bundles of first-order structures.
Our proof method is new even in the classical case of Polish groups, and is based entirely on formal algebraic properties of category quantifiers; in particular, we make no use of either metrizability or the strong Choquet game. Consequently, our proofs work equally well in the non-Hausdorff context, for open quasi-Polish groupoids, and more generally in the point-free context, for open localic groupoids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_06319 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Structural, point-free, non-Hausdorff topological realization of Borel groupoid actions Chen, Ruiyuan Logic Category Theory Dynamical Systems 03E15, 22A22, 22F10, 06D22 We extend the Becker--Kechris topological realization and change-of-topology theorems for Polish group actions in several directions. For Polish group actions, we prove a single result that implies the original Becker--Kechris theorems, as well as Sami's and Hjorth's sharpenings adapted levelwise to the Borel hierarchy; automatic continuity of Borel actions via homeomorphisms; and the equivalence of "potentially open" versus "orbitwise open" Borel sets. We also characterize "potentially open" $n$-ary relations, thus yielding a topological realization theorem for invariant Borel first-order structures. We then generalize to groupoid actions, and prove a result subsuming Lupini's Becker--Kechris-type theorems for open Polish groupoids, newly adapted to the Borel hierarchy, as well as topological realizations of actions on fiberwise topological bundles and bundles of first-order structures. Our proof method is new even in the classical case of Polish groups, and is based entirely on formal algebraic properties of category quantifiers; in particular, we make no use of either metrizability or the strong Choquet game. Consequently, our proofs work equally well in the non-Hausdorff context, for open quasi-Polish groupoids, and more generally in the point-free context, for open localic groupoids. |
| title | Structural, point-free, non-Hausdorff topological realization of Borel groupoid actions |
| topic | Logic Category Theory Dynamical Systems 03E15, 22A22, 22F10, 06D22 |
| url | https://arxiv.org/abs/2209.06319 |