Structurable equivalence relations and $\mathcal{L}_{ω_1ω}$ interpretations

Rishi Banerjee, Ruiyuan Chen
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Abstract

We show that the category of countable Borel equivalence relations (CBERs) is dually equivalent to the category of countable $\mathcal{L}_{\omega_1\omega}$ theories which admit a one-sorted interpretation of a particular theory we call $\mathcal{T}_\mathsf{LN} \sqcup \mathcal{T}_\mathsf{sep}$ that witnesses embeddability into $2^\mathbb{N}$ and the Lusin--Novikov uniformization theorem. This allows problems about Borel combinatorial structures on CBERs to be translated into syntactic definability problems in $\mathcal{L}_{\omega_1\omega}$, modulo the extra structure provided by $\mathcal{T}_\mathsf{LN} \sqcup \mathcal{T}_\mathsf{sep}$, thereby formalizing a folklore intuition in locally countable Borel combinatorics. We illustrate this with a catalogue of the precise interpretability relations between several standard classes of structures commonly used in Borel combinatorics, such as Feldman--Moore $\omega$-colorings and the Slaman--Steel marker lemma. We also generalize this correspondence to locally countable Borel groupoids and theories interpreting $\mathcal{T}_\mathsf{LN}$, which admit a characterization analogous to that of Hjorth--Kechris for essentially countable isomorphism relations.
可结构等价关系和 $mathcal{L}_{ω_1ω}$ 解释
我们证明,可数伯尔等价关系(CBERs)范畴与可数$\mathcal{L}_{\omega_1\omega}$理论范畴实际上是等价的,后者接受我们称之为$\mathcal{T}_\mathsf{LN}的特殊理论的单排序解释。我们称之为$\mathcal{T}_\mathsf{LN}{sqcup \mathcal{T}_\mathsf{sep}$理论的单排序解释,它见证了进入$2^\mathbb{N}$的可嵌入性以及卢辛--诺维科夫统一化定理。这使得关于 CBER 上的 Borel 组合结构的问题可以转化为$\mathcal{L}_{\omega_1\omega}$中的句法可定义性问题,并修改了$\mathcal{T}_\mathsf{LN}提供的额外结构。\sqcup)所提供的额外结构,从而将局部可数伯尔组合学中的一种民俗直觉形式化。我们列举了博尔组合学中常用的几类标准结构之间的精确可解释性关系,如费尔德曼--摩尔$\omega$着色和斯拉曼--斯泰尔标记稃,以此来说明这一点。我们还将这种对应关系推广到局部可数的伯尔群组和解释 $\mathcal{T}_\mathsf{LN}$ 的理论,这些理论的特征描述类似于 Hjorth--Kechris 对于本质上可数的同构关系的特征描述。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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