[Paper Review] On the structure of categorical abstract elementary classes with amalgamation

Monica VanDieren, Sebastien Vasey|arXiv (Cornell University)|Sep 4, 2015

Advanced Topology and Set Theory11 references3 citations

TL;DR

This paper establishes that categoricity in a sufficiently large cardinal implies key structural properties in abstract elementary classes (AECs) with amalgamation and no maximal models, including uniqueness of limit models, $μ$-saturation of the categoricity model, and the existence of a type-full good $μ$-frame. The core mechanism relies on proving symmetry for $μ$-splitting from the absence of the order property, enabling frame construction without assuming tameness or successor categoricity.

ABSTRACT

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 $μ\ge ext{LS} (K)$. If $K$ is categorical in a $λ\ge \beth_{\left(2^μ ight)^+}$, 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 $μ> ext{LS} (K)$, the model of size $λ$ is $μ$-saturated. 3) If $μ\ge \beth_{(2^{ ext{LS} (K)})^+}$ and $λ\ge \beth_{\left(2^{μ^+} ight)^+}$, 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.

Motivation & Objective

• To establish structural consequences of categoricity in high cardinals for AECs with amalgamation and no maximal models.
• To prove that categoricity in a $λ \geq \beth_{(2^\mu)^+}$ implies $μ$-saturation of the model of size $λ$, even without cofinality assumptions.
• To show that categoricity in a sufficiently large cardinal implies the existence of a type-full good $μ$-frame on saturated models in $\mathcal{K}_\mu$, without assuming tameness or successor categoricity.
• To demonstrate that symmetry of splitting, derived from the absence of the order property, is sufficient to construct good frames in AECs.
• To provide a framework for building local independence relations (good frames) directly from categoricity, bypassing traditional assumptions like tameness or successor cardinals.

Proposed method

• Utilizes the symmetry property of $μ$-splitting, previously isolated by VanDieren, as a central tool to derive structural properties.
• Proves that the failure of the order property implies symmetry of $μ$-splitting, establishing a key link between stability and independence in AECs.
• Applies the Shelah-Villaveces theorem to deduce $μ$-superstability from categoricity and amalgamation, enabling the use of stability-theoretic tools.
• Employs weak tameness (tameness over saturated models) as a substitute for full tameness in constructing good frames, which follows from categoricity in large cardinals.
• Constructs a type-full good $μ$-frame on the class of saturated models in $\mathcal{K}_\mu$ using symmetry, superstability, and weak tameness.
• Combines results on limit models and saturation to show that categoricity in a high enough cardinal implies structural coherence in the AEC.

Experimental results

Research questions

• RQ1 Does categoricity in a high enough cardinal imply the uniqueness of limit models in AECs with amalgamation and no maximal models?
• RQ2 Can $μ$-saturation of the categoricity model be derived without assuming cofinality conditions or tameness?
• RQ3 Is it possible to construct a good frame in an AEC with amalgamation and no maximal models using only categoricity and symmetry of splitting?
• RQ4 Can the existence of a good frame be established without assuming tameness or successor categoricity?
• RQ5 What is the role of the symmetry property of splitting in deriving structural results from categoricity in AECs?

Key findings

• Categoricity in a $λ \geq \beth_{(2^\mu)^+}$ implies that any two models $M_1$ and $M_2$ that are limit over a common $M_0$ in $\mathcal{K}_\mu$ are isomorphic over $M_0$, ensuring uniqueness of limit models.
• The model of size $\lambda$ is $\mu$-saturated whenever $\lambda \geq \beth_{(2^\mu)^+}$, even if $\operatorname{cf}(\lambda) \leq \mu$, showing that high categoricity cardinals behave as if they had large cofinality.
• If $\mu \geq \beth_{(2^{\operatorname{LS}(\mathcal{K})})^+}$ and $\lambda \geq \beth_{(2^{\mu^+})^+}$, then there exists a type-full good $\mu$-frame on the class of saturated models in $\mathcal{K}_\mu$.
• The existence of a good frame can be derived from categoricity in a large cardinal without assuming tameness, by using symmetry of splitting and weak tameness.
• Weak tameness (tameness over saturated models) follows from categoricity in a sufficiently large $\lambda$, enabling frame construction via stability-theoretic methods.
• The symmetry of $\mu$-splitting is equivalent to the continuity of the independence relation and can be derived from the absence of the order property, providing a key technical bridge to frame construction.