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Definability and Classification of Equivalence Relations and Logical Theories


Chen, Ruiyuan (2018) Definability and Classification of Equivalence Relations and Logical Theories. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/7BP3-VZ93.


This thesis consists of four independent papers.

In the first paper, joint with Kechris, we study the global aspects of structurability in the theory of countable Borel equivalence relations. For a class K of countable relational structures, a countable Borel equivalence relation E is said to be K-structurable if there is a Borel way to put a structure in K on each E-equivalence class. We show that K-structurability interacts well with various preorders commonly used in the classification of countable Borel equivalence relations. We consider the poset of classes of K-structurable equivalence relations for various K, under inclusion, and show that it is a distributive lattice. Finally, we consider the effect on K-structurability of various model-theoretic properties of K; in particular, we characterize the K such that every K-structurable equivalence relation is smooth.

In the second paper, we consider the classes of Kn-structurable equivalence relations, where Kn is the class of n-dimensional contractible simplicial complexes. We show that every Kn-structurable equivalence relation Borel embeds into one structurable by complexes in Kn with the further property that each vertex belongs to at most Mn := 2n-1(n2+3n+2)-2 edges; this generalizes a result of Jackson-Kechris-Louveau in the case n=1.

In the third paper, we consider the amalgamation property from model theory in an abstract categorical context. A category is said to have the amalgamation property if every pushout diagram has a cocone. We characterize the finitely generated categories I such that in every category with the amalgamation property, every I-shaped diagram has a cocone.

In the fourth paper, we prove a strong conceptual completeness theorem (in the sense of Makkai) for the infinitary logic Lω1ω: every countable Lω1ω-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 (L,T) and (L',T'), every Borel functor Mod(L',T') → Mod(L,T) between the respective groupoids of countable models is Borel naturally isomorphic to the functor induced by some L'ω1ω-interpretation of T in T', which generalizes a recent result of Harrison-Trainor, Miller, and Montalban in the case where T, T' are ℵ0-categorical.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Descriptive set theory; equivalence relations; model theory; categorical logic
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Awards:Scott Russell Johnson Graduate Dissertation Prize in Mathematics, 2018.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Kechris, Alexander S.
Thesis Committee:
  • Kechris, Alexander S. (chair)
  • Graber, Thomas B.
  • Lupini, Martino
  • Tamuz, Omer
Defense Date:9 May 2018
Funding AgencyGrant Number
Natural Sciences and Engineering Research Council of Canada (NSERC)Postgraduate Scholarship (PGS)
Record Number:CaltechTHESIS:05212018-182542598
Persistent URL:
Related URLs:
URLURL TypeDescription adapted for Chapter 2. adapted for Chapter 3. adapted for Chapter 4. adapted for Chapter 5.
Chen, Ruiyuan0000-0002-5891-8717
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:10920
Deposited By: Ruiyuan Chen
Deposited On:24 May 2018 23:27
Last Modified:04 Oct 2019 00:21

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