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 $\kappa$. We show: $\mathbf{Theorem}$ (The semantic-syntactic correspondence) An AEC $K$ is fully $(<\kappa)$-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 $\lambda \ge \text{LS}(K)$. * $K$ does not have the order property (defined in terms of Galois types). * There exist cardinals $\mu$ and $\lambda_0$ with $\mu \le \lambda_0 < \beth_{(2^{\text{LS}(K)})^+}$ such that $K$ is Galois stable in any $\lambda \ge \lambda_0$ with $\lambda = \lambda^{<\mu}$. $\mathbf{Theorem}$ Let $K$ be a fully $(<\kappa)$-tame and type short AEC with amalgamation, $\kappa = \beth_{\kappa} > \text{LS} (K)$. If $K$ is Galois stable, then the class of $\kappa$-Galois saturated models of $K$ admits an independence notion ($(<\kappa)$-coheir) which, except perhaps for extension, has the properties of forking in a first-order stable theory.Comment: 34 pages (v1 was split into this paper and arXiv:1503.01366

Universal classes near ℵ1\aleph_1

Shelah has provided sufficient conditions for an $L_{\omega_1, \omega}$-sentence $\psi$ to have arbitrarily large models and for a Morley-like theorem to hold of $\psi$. These conditions involve structural and set-theoretic assumptions on all the $\aleph_n$'s. Using tools of Boney, Shelah, and the second author, we give assumptions on $\aleph_0$ and $\aleph_1$ which suffice when $\psi$ is restricted to be universal: $\mathbf{Theorem}$ Assume $2^{\aleph_{0}} < 2 ^{\aleph_{1}}$. Let $\psi$ be a universal $L_{\omega_{1}, \omega}$-sentence. - If $\psi$ is categorical in $\aleph_{0}$ and $1 \leq I(\psi, \aleph_{1}) < 2 ^{\aleph_{1}}$, then $\psi$ has arbitrarily large models and categoricity of $\psi$ in some uncountable cardinal implies categoricity of $\psi$ in all uncountable cardinals. - If $\psi$ is categorical in $\aleph_1$, then $\psi$ is categorical in all uncountable cardinals. The theorem generalizes to the framework of $L_{\omega_1, \omega}$-definable tame abstract elementary classes with primes.Comment: 12 pages; Corrected typos; Rewrote part of the introductio

Categoricity in multiuniversal classes

The third author has shown that Shelah's eventual categoricity conjecture holds in universal classes: class of structures closed under isomorphisms, substructures, and unions of chains. We extend this result to the framework of multiuniversal classes. Roughly speaking, these are classes with a closure operator that is essentially algebraic closure (instead of, in the universal case, being essentially definable closure). Along the way, we prove in particular that Galois (orbital) types in multiuniversal classes are determined by their finite restrictions, generalizing a result of the second author.Comment: 15 page

Universal abstract elementary classes and locally multipresentable categories

We exhibit an equivalence between the model-theoretic framework of universal classes and the category-theoretic framework of locally multipresentable categories. We similarly give an equivalence between abstract elementary classes (AECs) admitting intersections and locally polypresentable categories. We use these results to shed light on Shelah's presentation theorem for AECs.Comment: 14 pages. Some typos remove

Saturation and solvability in abstract elementary classes with amalgamation

$\mathbf{Theorem.}$ Let $K$ be an abstract elementary class (AEC) with amalgamation and no maximal models. Let $\lambda > \text{LS} (K)$. If $K$ is categorical in $\lambda$, then the model of cardinality $\lambda$ is Galois-saturated. This answers a question asked independently by Baldwin and Shelah. We deduce several corollaries: $K$ has a unique limit model in each cardinal below $\lambda$, (when $\lambda$ is big-enough) $K$ is weakly tame below $\lambda$, and the thresholds of several existing categoricity transfers can be improved. We also prove a downward transfer of solvability (a version of superstability introduced by Shelah): $\mathbf{Corollary.}$ Let $K$ be an AEC with amalgamation and no maximal models. Let $\lambda > \mu > \text{LS} (K)$. If $K$ is solvable in $\lambda$, then $K$ is solvable in $\mu$.Comment: 19 page

Hanf Numbers and Presentation Theorems in AECs

We prove that a strongly compact cardinal is an upper bound for a Hanf number for amalgamation, etc. in AECs using both semantic and syntactic methods. To syntactically prove non-disjoint amalgamation, a different presentation theorem than Shelah's is needed. This relational presentation theorem has the added advantage of being {\it functorial}, which allows the transfer of amalgamation

On universal modules with pure embeddings

We show that certain classes of modules have universal models with respect to pure embeddings. $Theorem.$ Let $R$ be a ring, $T$ a first-order theory with an infinite model extending the theory of $R$-modules and $K^T=(Mod(T), \leq_{pp})$ (where $\leq_{pp}$ stands for pure submodule). Assume $K^T$ has joint embedding and amalgamation. If $\lambda^{|T|}=\lambda$ or $\forall \mu < \lambda( \mu^{|T|} < \lambda)$, then $K^T$ has a universal model of cardinality $\lambda$. As a special case we get a recent result of Shelah [Sh17, 1.2] concerning the existence of universal reduced torsion-free abelian groups with respect to pure embeddings. We begin the study of limit models for classes of $R$-modules with joint embedding and amalgamation. We show that limit models with chains of long cofinality are pure-injective and we characterize limit models with chains of countable cofinality. This can be used to answer Question 4.25 of [Maz]. As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both.Comment: 17 page

Metric abstract elementary classes as accessible categories

We show that metric abstract elementary classes (mAECs) are, in the sense of [LR] (i.e. arXiv:1404.2528), coherent accessible categories with directed colimits, with concrete $\aleph_1$-directed colimits and concrete monomorphisms. More broadly, we define a notion of $\kappa$-concrete AEC---an AEC-like category in which only the $\kappa$-directed colimits need be concrete---and develop the theory of such categories, beginning with a category-theoretic analogue of Shelah's Presentation Theorem and a proof of the existence of an Ehrenfeucht-Mostowski functor in case the category is large. For mAECs in particular, arguments refining those in [LR] yield a proof that any categorical mAEC is $\mu$-d-stable in many cardinals below the categoricity cardinal.Comment: v2: changed terminology. v3: tightened inequalities. v4: clarifying notes added. v5: referee's comments incorporated, with substantial improvement

On categoricity in successive cardinals

We investigate, in ZFC, the behavior of abstract elementary classes (AECs) categorical in many successive small cardinals. We prove for example that a universal $\mathbb{L}_{\omega_1, \omega}$ sentence categorical on an end segment of cardinals below $\beth_\omega$ must be categorical also everywhere above $\beth_\omega$. This is done without any additional model-theoretic hypotheses (such as amalgamation or arbitrarily large models) and generalizes to the much broader framework of tame AECs with weak amalgamation and coherent sequences.Comment: 19 page

Equivalent definitions of superstability in tame abstract elementary classes

In the context of abstract elementary classes (AECs) with a monster model, several possible definitions of superstability have appeared in the literature. Among them are no long splitting chains, uniqueness of limit models, and solvability. Under the assumption that the class is tame and stable, we show that (asymptotically) no long splitting chains implies solvability and uniqueness of limit models implies no long splitting chains. Using known implications, we can then conclude that all the previously-mentioned definitions (and more) are equivalent: $\mathbf{Corollary}$ Let $K$ be a tame AEC with a monster model. Assume that $K$ is stable in a proper class of cardinals. The following are equivalent: 1) For all high-enough $\lambda$, $K$ has no long splitting chains. 2) For all high-enough $\lambda$, there exists a good $\lambda$-frame on a skeleton of $K_\lambda$. 3) For all high-enough $\lambda$, $K$ has a unique limit model of cardinality $\lambda$. 4) For all high-enough $\lambda$, $K$ has a superlimit model of cardinality $\lambda$. 5) For all high-enough $\lambda$, the union of any increasing chain of $\lambda$-saturated models is $\lambda$-saturated. 6) There exists $\mu$ such that for all high-enough $\lambda$, $K$ is $(\lambda, \mu)$-solvable. This gives evidence that there is a clear notion of superstability in the framework of tame AECs with a monster model.Comment: 24 page