arXiv subjects

Ruiyuan Chen

Publications and source records attributed to Ruiyuan Chen.

25 records · Page 2Linked to original sources

arXiv2020

Borel and analytic sets in locales

We systematically develop analogs of basic concepts from classical descriptive set theory in the context of pointless topology. Our starting point is to take the elements of the free complete Boolean algebra generated by the frame $\mathcal{O}(X)$ of opens to be the "$\infty$-Borel sets" in a locale $X$. We show that several known results in locale theory may be interpreted in this framework as direct analogs of classical descriptive set-theoretic facts, including e.g., the Lusin separation, Lusin-Suslin, and Baire category theorems for locales; we also prove several extensions of these results. We give a detailed analysis of various notions of image, and prove that a continuous map need not have an $\infty$-Borel image. We introduce the category of "analytic $\infty$-Borel locales" as the regular completion under images of the unary site of locales and $\infty$-Borel maps, and prove analogs of several classical results about analytic sets, such as a boundedness theorem for well-founded analytic relations. We also consider the "positive $\infty$-Borel sets" of a locale, formed from opens without using $\neg$. We in fact work throughout with $κ$-copresented $κ$-locales and $κ$-Borel sets for arbitrary regular $ω_1 \le κ\le \infty$, thereby incorporating the classical context as the special case $κ= ω_1$. The basis for the aforementioned localic results is a detailed study of various known and new methods for presenting $κ$-frames, $κ$-Boolean algebras, etc. In particular, we introduce a new type of "two-sided posite" for presenting $(κ, κ)$-frames $A$ (i.e., both $A$ and $A^{op}$ are $κ$-frames), and use this to prove a general $κ$-ary interpolation theorem for $(κ, κ)$-frames, which dualizes to the aforementioned separation theorems.

math.LO↗

arXiv2019

Representing Polish groupoids via metric structures

We prove that every open $σ$-locally Polish groupoid $G$ is Borel equivalent to the groupoid of models on the Urysohn sphere $\mathbb{U}$ of an $\mathcal{L}_{ω_1ω}$-sentence in continuous logic. In particular, the orbit equivalence relations of such groupoids are up to Borel bireducibility precisely those of Polish group actions, answering a question of Lupini. Analogously, every non-Archimedean (i.e., every unit morphism has a neighborhood basis of open subgroupoids) open quasi-Polish groupoid is Borel equivalent to the groupoid of models on $\mathbb{N}$ of an $\mathcal{L}_{ω_1ω}$-sentence in discrete logic. The proof in fact gives a topological representation of $G$ as the groupoid of isomorphisms between a "continuously varying" family of structures over the space of objects of $G$, constructed via a topological Yoneda-like lemma of Moerdijk for localic groupoids and its metric analog. Other ingredients in our proof include the Lopez-Escobar theorem for continuous logic, a uniformization result for full Borel functors between open quasi-Polish groupoids, a uniform Borel version of Katětov's construction of $\mathbb{U}$, groupoid versions of the Pettis and Birkhoff--Kakutani theorems, and a development of the theory of non-Hausdorff topometric spaces and their quotients.

math.LO↗

arXiv2019

Borel functors, interpretations, and strong conceptual completeness for $\mathcal L_{ω_1ω}$

We prove a strong conceptual completeness theorem (in the sense of Makkai) for the infinitary logic $\mathcal L_{ω_1ω}$: every countable $\mathcal L_{ω_1ω}$-theory can be canonically recovered from its standard Borel groupoid of countable models, up to a suitable syntactical notion of equivalence. This implies that given two theories $(\mathcal L, \mathcal T)$ and $(\mathcal L', \mathcal T')$ (in possibly different languages $\mathcal L, \mathcal L'$), every Borel functor $\mathsf{Mod}(\mathcal L', \mathcal T') \to \mathsf{Mod}(\mathcal L, \mathcal T)$ between the respective groupoids of countable models is Borel naturally isomorphic to the functor induced by some $\mathcal L'_{ω_1ω}$-interpretation of $\mathcal T$ in $\mathcal T'$. This generalizes a recent result of Harrison-Trainor, Miller, and Montalbán in the case where $\mathcal T, \mathcal T'$ each have a single countable model up to isomorphism.

math.LO↗

arXiv2019

Bounds on Continuous Scott Rank

An analog of Nadel's effective bound for the continuous Scott rank of metric structures, developed by Ben Yaacov, Doucha, Nies, and Tsankov, will be established: Let $\mathscr{L}$ be a language of continuous logic with code $\hat{\mathscr{L}}$. Let $Ω$ be a weak modulus of uniform continuity with code $\hatΩ$. Let $\mathcal{D}$ be a countable $\mathscr{L}$-pre-structure. Let $\bar{\mathcal{D}}$ denote the completion structure of $\mathcal{D}$. Then $\mathrm{SR}_Ω(\bar{D}) \leq ω_1^{\hat{\mathscr{L}}\oplus\hatΩ\oplus\mathcal{D}}$, the Church-Kleene ordinal relative to $\hat{\mathscr{L}}\oplus\hatΩ\oplus\mathcal{D}$.

math.LO↗

arXiv2018

Structurable equivalence relations

For a class $\mathcal K$ of countable relational structures, a countable Borel equivalence relation $E$ is said to be $\mathcal K$-structurable if there is a Borel way to put a structure in $\mathcal K$ on each $E$-equivalence class. We study in this paper the global structure of the classes of $\mathcal K$-structurable equivalence relations for various $\mathcal K$. We show that $\mathcal K$-structurability interacts well with several kinds of Borel homomorphisms and reductions commonly used in the classification of countable Borel equivalence relations. We consider the poset of classes of $\mathcal K$-structurable equivalence relations for various $\mathcal K$, under inclusion, and show that it is a distributive lattice; this implies that the Borel reducibility preordering among countable Borel equivalence relations contains a large sublattice. Finally, we consider the effect on $\mathcal K$-structurability of various model-theoretic properties of $\mathcal K$. In particular, we characterize the $\mathcal K$ such that every $\mathcal K$-structurable equivalence relation is smooth, answering a question of Marks.

math.LO↗

arXiv2017

Borel structurability by locally finite simplicial complexes

We show that every countable Borel equivalence relation structurable by $n$-dimensional contractible simplicial complexes embeds into one which is structurable by such complexes with the further property that each vertex belongs to at most $M_n := 2^{n-1}(n^2+3n+2)-2$ edges; this generalizes a result of Jackson-Kechris-Louveau in the case $n = 1$. The proof is based on that of a classical result of Whitehead on countable CW-complexes.

math.LO↗

arXiv2016

Amalgamable diagram shapes

A category has the amalgamation property (AP) if every pushout diagram has a cocone, and the joint embedding property (JEP) if every finite coproduct diagram has a cocone. We show that for a finitely generated category $\mathbf I$, the following are equivalent: (i) every $\mathbf I$-shaped diagram in a category with the AP and the JEP has a cocone; (ii) every $\mathbf I$-shaped diagram in the category of sets and injections has a cocone; (iii) a certain canonically defined category $\mathcal{L}(\mathbf{I})$ of "paths" in $\mathbf I$ has only idempotent endomorphisms. When $\mathbf I$ is a finite poset, these are further equivalent to: (iv) every upward-closed subset of $\mathbf I$ is simply-connected; (v) $\mathbf I$ can be built inductively via some simple rules. Our proof also shows that these conditions are decidable for finite $\mathbf I$.

math.CT↗