We develop a notion of forking for Galois-types in the context of Abstract Elementary Classes (AECs). Under the hypotheses that an AEC $K$ is tame, type-short, and failure of an order-property, we consider
{\bf Definition.} Let $M_0 \prec N$ be models from $K$ and $A$ be a set. We say that the Galois-type of $A$ over $M$ \emph{does not fork over $M_0$} iff for all small $a \in A$ and all small $N^- \prec N$, we have that Galois-type of $a$ over $N^-$ is realized in $M_0$.
Assuming property (E) (see Definition 3.3) we show that this non-forking is a well behaved notion of independence, in particular satisfies symmetry and uniqueness and has a corresponding U-rank. We find conditions for a universal local character, in particular derive superstability-like property from little more than categoricity in a \big cardinal". Finally, we show that under large cardinal axioms the proofs are simpler and the non-forking is more powerful. In [BGKV] it is established that this notion of non-forking is the only independence relation possible.
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.
We show, assuming a mild set-theoretic hypothesis, that if an abstract elementary class (AEC) has a superstable-like forking notion for models of cardinality $λ$ and a superstable-like forking notion for models of cardinality $λ^+$, then orbital types over models of cardinality $λ^+$ are determined by their restrictions to submodels of cardinality $λ$. By a superstable-like forking notion, we mean here a good frame, a central concept of Shelah's book on AECs. It is known that locality of orbital types together with the existence of a superstable-like notion for models of cardinality $λ$ implie…
We combine two approaches to the study of classification theory of AECs: 1. that of Shelah: studying non-forking frames without assuming the amalgamation property but assuming the existence of uniqueness triples and 2. that of Grossberg and VanDieren: (studying non-splitting) assuming the amalgamation property and tameness. In [JrSh875], we derive a good non-forking $λ^+$-frame from a semi-good non-forking $λ$-frame. But the classes $K_{λ^+}$ and $\preceq \restriction K_{λ^+}$ are replaced: $K_{λ^+}$ is restricted to the saturated models and the partial order $\preceq \restriction K_{λ^+}$ is …
We introduce the notion of a w-good $λ$-frame which is a weakening of Shelah's notion of a good $λ$-frame. Existence of a w-good $λ$-frame implies existence of a model of size $λ^{++}$. Tameness and amalgamation imply extension of a w-good $λ$-frame to larger models. As an application we show: $\textbf{Theorem}$ Suppose $2^λ< 2^{λ^{+}} < 2^{λ^{++}}$ and $2^{λ^{+}} > λ^{++}$. If $I(K, λ) = I(K, λ^{+}) = 1 \leq I(K, λ^{++}) < 2^{λ^{++}}$ and $K$ is $(λ, λ^+)$-tame, then $K_{λ^{+++}} \neq \emptyset$. The proof presented clarifies some of the details of the main theorem of [Sh576] and avoids using…
In the setup of abstract elementary classes satisfying a local version of superstability, we prove the uniqueness property for $μ$-forking, a certain independence notion arising from splitting. This had been a longstanding technical difficulty when constructing forking-like notions in this setup. As an application, we show that the two versions of forking symmetry appearing in the literature (the one defined by Shelah for good frames and the one defined by VanDieren for splitting) are equivalent.
In this paper, we study a stability transfer theorem in d-tame Metric Abstract Elementary classes, in a similar way as in [BaKuVa], but using superstability-like assumptions which involves a new independence notion (Tame Independence) instead of $\aleph_0$-locality.
Forking is a central notion of model theory, generalizing linear independence in vector spaces and algebraic independence in fields. We develop the theory of forking in abstract, category-theoretic terms, for reasons both practical (we require a characterization suitable for work in $μ$-abstract elementary classes, i.e. accessible categories with all morphisms monomorphisms) and expository (we hope, with this account, to make forking accessible - and useful - to a broader mathematical audience). In particular, we present an axiomatic definition of what we call a stable independence notion on a…
Grossberg and VanDieren have started a program to develop a stability theory for tame classes. We prove, for instance, that for tame abstract elementary classes satisfying the amlagamation property and for large enough cardinals kappa, stability in kappa implies stability in kappa^{+n} for each natural number n.
We study general methods to build forking-like notions in the framework of tame abstract elementary classes (AECs) with amalgamation. We show that whenever such classes are categorical in a high-enough cardinal, they admit a good frame: a forking-like notion for types of singleton elements. $\mathbf{Theorem}$ (Superstability from categoricity) Let $K$ be a $(<κ)$-tame AEC with amalgamation. If $κ= \beth_κ> \text{LS} (K)$ and $K$ is categorical in a $λ> κ$, then: * $K$ is stable in all cardinals $\ge κ$. * $K$ is categorical in $κ$. * There is a type-full good $λ$-frame with underlying class $K…
Let K be an abstract elementary classes which has arbitrarily large models and satisfies the amalgamation and joint embedding properties. Theorem 1. Suppose K is χ-tame. If K is categorical in some λ^+ >LS(K) then it is categorical in all μ\geq (λ+χ)^+. Theorem 2. If K is LS(K)-tame and is categorical both in LS(K) and in LS(K)^+ then K is categorical in all μ\geq LS(K).
We introduce tame abstract elementary classes as a generalization of all cases of abstract elementary classes that are known to permit development of stability-like theory. In this paper we explore stability results in this context. We assume that $\K$ is a tame abstract elementary class satisfying the amalgamation property with no maximal model. The main results include: (1) Galois-stability above the Hanf number implies that κ(K) is less than the Hanf number. Where κ(K) is the parallel of \kapppa(T) for f.o. T. (2) We use (1) to construct Morley sequences (for non-splitting) improving previo…
Let K be an Abstract Elemenetary Class satisfying the amalgamation and the joint embedding property, let μbe the Hanf number of K. Suppose K is tame. MAIN COROLLARY: (ZFC) If K is categorical in a successor cardinal bigger than \beth_{(2^μ)^+} then K is categorical in all cardinals greater than \beth_{(2^μ)^+}. This is an improvment of a Theorem of Makkai and Shelah ([Sh285] who used a strongly compact cardinal for the same conclusion) and Shelah's downward categoricity theorem for AECs with amalgamation (from [Sh394]).
Tame abstract elementary classes are a broad nonelementary framework for model theory that encompasses several examples of interest. In recent years, progress toward developing a classification theory for them have been made. Abstract independence relations such as Shelah's good frames have been found to be key objects. Several new categoricity transfers have been obtained. We survey these developments using the following result (due to the second author) as our guiding thread: $\mathbf{Theorem}$ If a universal class is categorical in cardinals of arbitrarily high cofinality, then it is catego…
In this paper we address a problem posed by Shelah in 1999 to find a suitable notion for superstability for abstract elementary classes in which limit models of cardinality $μ$ are saturated. Theorem 1. Suppose that $\mathcal{K}$ is a $χ$-tame abstract elementary class with no maximal models satisfying the joint embedding property and the amalgamation property. Suppose $μ$ is a cardinal with $μ\geq\beth_{(2^{LS(\mathcal{K})+χ})^+}$. Let $M$ be a model of cardinality $μ$. If $\mathcal{K}$ is both $χ$-stable and $μ$-stable and satisfies the $μ$-superstability assumptions, then any two $μ$-limit …
We introduce a family of rank functions and related notions of total transcendence for Galois types in abstract elementary classes. We focus, in particular, on abstract elementary classes satisfying the condition know as tameness (currently suspected to be a necessary condition for the development of a reasonable classification theory) where the connections between stability and total transcendence are most evident. As a byproduct, we obtain a partial upward stability transfer result for tame abstract elementary classes stable in a cardinal lambda satisfying lambda^{aleph_0}, a substantial gen…
The assumption that an AEC is tame is a powerful assumption permitting development of stability theory for AECs with the amalgamation property. Lately several upward categoricity theorems were discovered where tameness replaces strong set-theoretic assumptions. We present in this article two sufficient conditions for tameness, both in form of strong amalgamation properties that occur in nature. One of them was used recently to prove that several Hrushovski classes are tame. This is done by introducing the property of weak $(μ,n)$-uniqueness which makes sense for all AECs (unlike Shelah's origi…
We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness (a locality property for orbital types), and are stable (in terms of the number of orbital types) in some cardinal. Assuming the singular cardinal hypothesis (SCH), we prove a full characterization of the (high-enough) stability cardinals, and connect the stability spectrum with the behavior of saturated models. We deduce (in ZFC) that if a class is stable on a tail of cardinals, then it has no long splitting chains (the converse is known). This indicates that there is a clear not…
We show that various tameness assertions about abstract elementary classes imply the existence of large cardinals under mild cardinal arithmetic assumptions.
We study the problem of extending an abstract independence notion for types of singletons (what Shelah calls a good frame) to longer types. Working in the framework of tame abstract elementary classes, we show that good frames can always be extended to types of independent sequences. As an application, we show that tameness and a good frame imply Shelah's notion of dimension is well-behaved, complementing previous work of Jarden and Sitton. We also improve a result of the first author on extending a frame to larger models.
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 …
Based on Crapo's theory of one point extensions of combinatorial geometries, we find various classes of geometric lattices that behave very well from the point of view of stability theory. One of them, $(\mathbf{K}^3, \preccurlyeq)$, is $ω$-stable, it has a monster model and an independence calculus that satisfies all the usual properties of non-forking. On the other hand, these classes are rather unusual, e.g. in $(\mathbf{K}^3, \preccurlyeq)$ the Smoothness Axiom fails, and so $(\mathbf{K}^3, \preccurlyeq)$ is not an $\mathrm{AEC}$.
Let K be an Abstract Elementary Class. Under the asusmptions that K has a nicely behaved forking-like notion, regular types and existence of some prime models we establish a decomposition theorem for such classes. The decomposition implies a main gap result for the class K. The setting is general enough to cover \aleph_0-stable first-order theories (proved by Shelah in 1982), Excellent Classes of atomic models of a first order tehory (proved Grossberg and Hart 1987) and the class of submodels of a large sequentially homogenuus \aleph_0-stable model (which is new).
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…
We broaden the framework of metric abstract elementary classes (mAECs) in several essential ways, chiefly by allowing the metric to take values in a well-behaved quantale. As a proof of concept we show that the result of Boney and Zambrano on (metric) tameness under a large cardinal assumption holds in this more general context. We briefly consider a further generalization to partial metric spaces, and hint at connections to classes of fuzzy structures, and structures on sheaves.
This paper is part of a program initiated by Saharon Shelah to extend the model theory of first order logic to the non-elementary setting of abstract elementary classes (AECs). An abstract elementary class is a semantic generalization of the class of models of a complete first order theory with the elementary substructure relation. We examine the symmetry property of splitting (previously isolated by the first author) in AECs with amalgamation that satisfy a local definition of superstability. The key results are a downward transfer of symmetry and a deduction of symmetry from failure of the o…
We study classes of right-angled Coxeter groups with respect to the strong submodel relation of parabolic subgroup. We show that the class of all right-angled Coxeter group is not smooth, and establish some general combinatorial criteria for such classes to be abstract elementary classes, for them to be finitary, and for them to be tame. We further prove two combinatorial conditions ensuring the strong rigidity of a right-angled Coxeter group of arbitrary rank. The combination of these results translate into a machinery to build concrete examples of $\mathrm{AECs}$ satisfying given model-theor…
$\mathbf{Theorem.}$ Let $K$ be an abstract elementary class (AEC) with amalgamation and no maximal models. Let $λ> \text{LS} (K)$. If $K$ is categorical in $λ$, then the model of cardinality $λ$ 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 $λ$, (when $λ$ is big-enough) $K$ is weakly tame below $λ$, 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 She…
Nash structure of curves over a valued field and tame henselian rationality
Sorbonne Université, Université Paris-Diderot, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG（索邦大学，巴黎第七大学，法国国家科学研究中心，儒勒·让苏-巴黎左岸数学研究所）；Sorbonne Université, Institut de Mathématiques de Jussieu-Paris Rive Gauche, CNRS（索邦大学，儒勒·让苏-巴黎左岸数学研究所，法国国家科学研究中心）
Our main result (Theorem 1) suggests a possible dividing line ($μ$-superstable $+$ $μ$-symmetric) for abstract elementary classes without using extra set-theoretic assumptions or tameness. This theorem illuminates the structural side of such a dividing line. Theoerem 1: Let $\mathcal{K}$ be an abstract elementary class with no maximal models of cardinality $μ^+$ which satisfies the joint embedding and amalgamation properties. Suppose $μ\geq LS(\mathcal{K})$. If $\mathcal{K}$ is $μ$- and $μ^+$-superstable and satisfies $μ^+$-symmetry, then for any increasing sequence $\langle M_i\in\mathcal{K}_…
In this paper, we apply results of \cite{Va3} and use towers to transfer symmetry from $μ^+$ down to $μ$ in superstable abstract elementary classes without using extra set-theoretic assumptions or tameness. Theorem. Suppose $\mathcal{K}$ is an abstract elementary class satisfying the amalgamation and joint embedding properties and that $\mathcal{K}$ is both $μ$- and $μ^+$-superstable. If $\mathcal{K}$ has symmetry for non-$μ^+$-splitting, then $\mathcal{K}$ has symmetry for non-$μ$-splitting. This is a new application of towers which were introduced by Shelah and Villaveces \cite{ShVi} and lat…
Let K be an abstract elementary class satisfying the joint embedding and the amalgamation properties. Let m be a cardinal above the the Löwenheim-Skolem number of the class. Suppose K satisfies the disjoint amalgamation property for limit models of cardinality m. If K is m-Galois-stable, has no m-Vaughtian Pairs, does not have long splitting chains, and satisfies locality of splitting, for the precise description of long splitting chains and locality}, then any two (m,sigma_i)-limits over M for (i in {1,2}) are isomorphic over M. This theorem extends results of Shelah, Kolman and Shelah, and S…
Let ${\bf K}$ be an $\mathrm{LS}({\bf K})$-short abstract elementary class and assume more than the existence of a monster model (amalgamation over sets and arbitrarily large models). Suppose ${\bf K}$ is categorical in some $μ>\mathrm{LS}({\bf K})$, then it is categorical in all $μ'\geqμ$. Our result removes the successor requirement of $μ$ made by Grossberg-VanDieren, at the cost of using shortness instead of tameness; and of using amalgamation over sets instead of over models. It also removes the primes requirement by Vasey which assumes tameness and amalgamation over models. As a corollary…
We present a way of topologizing sets of Galois types over structures in abstract elementary classes with amalgamation. In the elementary case, the topologies thus produced refine the syntactic topologies familiar from first order logic. We exhibit a number of natural correspondences between the model-theoretic properties of classes and their constituent models and the topological properties of the associated spaces. Tameness of Galois types, in particular, emerges as a topological separation principle.
We show that a number of results on abstract elementary classes (AECs) hold in accessible categories with concrete directed colimits. In particular, we prove a generalization of a recent result of Boney on tameness under a large cardinal assumption. We also show that such categories support a robust version of the Ehrenfeucht-Mostowski construction. This analysis has the added benefit of producing a purely language-free characterization of AECs, and highlights the precise role played by the coherence axiom.
We ask, when is a property of a model a logical property? According to the so-called Tarski-Sher criterion this is the case when the property is preserved by isomorphisms. We relate this to model-theoretic characteristics of abstract logics in which the model class is definable. This results in a graded concept of logicality in the terminology of Sagi. We investigate which characteristics of logics, such as variants of the Löwenheim-Skolem Theorem, Completeness Theorem, and absoluteness, are relevant from the logicality point of view, continuing earlier work by Bonnay, Feferman, and Sagi. We s…
We prove: Main Theorem: Let $\mathcal{K}$ be an abstract elementary class satisfying the joint embedding and the amalgamation properties with no maximal models of cardinality $μ$. Let $μ$ be a cardinal above the the Löwenheim-Skolem number of the class. If $\mathcal{K}$ is $μ$-Galois-stable, has no $μ$-Vaughtian Pairs, does not have long splitting chains, and satisfies locality of splitting, then any two $(μ,σ_\ell)$-limits over $M$, for $\ell\in\{1,2\}$, are isomorphic over $M$. This theorem extends results of Shelah from \cite{Sh394}, \cite{Sh576}, \cite{Sh600}, Kolman and Shelah in \cite{Ko…
