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Sebastien Vasey

Publications and source records attributed to Sebastien Vasey.

44 records · Page 3Linked to original sources
arXiv2016

Canonical forking in AECs

Boney and Grossberg [BG] proved that every nice AEC has an independence relation. We prove that this relation is unique: In any given AEC, there can exist at most one independence relation that satisfies existence, extension, uniqueness and local character. While doing this, we study more generally properties of independence relations for AECs and also prove a canonicity result for Shelah's good frames. The usual tools of first-order logic (like the finite equivalence relation theorem or the type amalgamation theorem in simple theories) are not available in this context. In addition to the loss of the compactness theorem, we have the added difficulty of not being able to assume that types are sets of formulas. We work axiomatically and develop new tools to understand this general framework.

math.LO↗
arXiv2016

Infinitary stability theory

We introduce a new device in the study of abstract elementary classes (AECs): Galois Morleyization, which consists in expanding the models of the class with a relation for every Galois type of length less than a fixed cardinal $κ$. We show: $\mathbf{Theorem}$ (The semantic-syntactic correspondence) An AEC $K$ is fully $(<κ)$-tame and type short if and only if Galois types are syntactic in the Galois Morleyization. This exhibits a correspondence between AECs and the syntactic framework of stability theory inside a model. We use the correspondence to make progress on the stability theory of tame and type short AECs. The main theorems are: $\mathbf{Theorem}$ Let $K$ be a $\text{LS}(K)$-tame AEC with amalgamation. The following are equivalent: * $K$ is Galois stable in some $λ\ge \text{LS}(K)$. * $K$ does not have the order property (defined in terms of Galois types). * There exist cardinals $μ$ and $λ_0$ with $μ\le λ_0 < \beth_{(2^{\text{LS}(K)})^+}$ such that $K$ is Galois stable in any $λ\ge λ_0$ with $λ= λ^{<μ}$. $\mathbf{Theorem}$ Let $K$ be a fully $(<κ)$-tame and type short AEC with amalgamation, $κ= \beth_κ > \text{LS} (K)$. If $K$ is Galois stable, then the class of $κ$-Galois saturated models of $K$ admits an independence notion ($(<κ)$-coheir) which, except perhaps for extension, has the properties of forking in a first-order stable theory.

math.LO↗
arXiv2016

On the structure of categorical abstract elementary classes with amalgamation

For $K$ an abstract elementary class with amalgamation and no maximal models, we show that categoricity in a high-enough cardinal implies structural properties such as the uniqueness of limit models and the existence of good frames. This improves several classical results of Shelah. $\mathbf{Theorem}$ Let $μ\ge \text{LS} (K)$. If $K$ is categorical in a $λ\ge \beth_{\left(2^μ\right)^+}$, then: 1) Whenever $M_0, M_1, M_2 \in K_μ$ are such that $M_1$ and $M_2$ are limit over $M_0$, we have $M_1 \cong_{M_0} M_2$. 2) If $μ> \text{LS} (K)$, the model of size $λ$ is $μ$-saturated. 3) If $μ\ge \beth_{(2^{\text{LS} (K)})^+}$ and $λ\ge \beth_{\left(2^{μ^+}\right)^+}$, then there exists a type-full good $μ$-frame with underlying class the saturated models in $K_μ$. Our main tool is the symmetry property of splitting (previously isolated by the first author). The key lemma deduces symmetry from failure of the order property.

math.LO↗
arXiv2016

$μ$-Abstract Elementary Classes and other generalizations

We introduce $μ$-Abstract Elementary Classes ($μ$-AECs) as a broad framework for model theory that includes complete boolean algebras and Dirichlet series, and begin to develop their classification theory. Moreover, we note that $μ$-AECs correspond precisely to accessible categories in which all morphisms are monomorphisms, and begin the process of reconciling these divergent perspectives: not least, the preliminary classification-theoretic results for μ-AECs transfer directly to accessible categories with monomorphisms.

math.LO↗
arXiv2015

Categoricity and infinitary logics

We point out a gap in Shelah's proof of the following result: $\mathbf{Claim}$ Let $K$ be an abstract elementary class categorical in unboundedly many cardinals. Then there exists a cardinal $λ$ such that whenever $M, N \in K$ have size at least $λ$, $M \le N$ if and only if $M \preceq_{L_{\infty, \text{LS} (K)^+}} N$. The importance of the claim lies in the following theorem, implicit in Shelah's work: $\mathbf{Theorem}$ Assume the claim. Let $K$ be an abstract elementary class categorical in unboundedly many cardinals. Then the class of $λ$ such that: 1) $K$ is categorical in $λ$; 2) $K$ has amalgamation in $λ$; and 3) there is a good $λ$-frame with underlying class $K_λ$ is stationary. We give a proof and discuss some related questions.

math.LO↗
arXiv2015

Forking and superstability in tame AECs

We prove that any tame abstract elementary class categorical in a suitable cardinal has an eventually global good frame: a forking-like notion defined on all types of single elements. This gives the first known general construction of a good frame in ZFC. We show that we already obtain a well-behaved independence relation assuming only a superstability-like hypothesis instead of categoricity. These methods are applied to obtain an upward stability transfer theorem from categoricity and tameness, as well as new conditions for uniqueness of limit models.

math.LO↗
arXiv2014

Indiscernible extraction and Morley sequences

We present a new proof of the existence of Morley sequences in simple theories. We avoid using the Erdős-Rado theorem and instead use only Ramsey's theorem and compactness. The proof shows that the basic theory of forking in simple theories can be developed using only principles from "ordinary mathematics", answering a question of Grossberg, Iovino and Lessmann, as well as a question of Baldwin.

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