Sebastien Vasey, Indiscernible extraction and Morley sequences, Notre Dame Journal of Formal Logic 58 (2017), no. 1, 127–132. Publisher version pdf (see all versions) arXiv.
In simple theories, Morley sequences can be built using only Ramsey's theorem and compactness. This shows that the basic theory of forking in simple theories can be developped using only principles from "elementary mathematics" and answers questions of Grossberg-Iovino-Lessmann and Baldwin. Using an argument of Itay Kaplan, we also obtain a new characterization of simple theories in terms of a property of forking we call dual finite character.
Will Boney, Rami Grossberg, Alexei Kolesnikov, and Sebastien Vasey, Canonical forking in AECs, Annals of Pure and Applied Logic 167 (2016), no. 7, 590–613. Publisher version pdf (see all versions) arXiv.
An abstract elementary class can have at most one forking-like notion. This extends the well-known canonicity proof of Harnik and Harrington but the methods are different: we do not rely on the compactness theorem and work with Galois (orbital) types. Along the way, we study relationship between the abstract properties of independence. We give an axiomatic proof that symmetry follows from no order property. This is used in subsequent papers to build a good frame (a local forking-like notion in AECs).
Sebastien Vasey, Forking and superstability in tame AECs, The Journal of Symbolic Logic 81 (2016), no. 1, 357–383. Publisher version pdf (see all versions) arXiv.
Any tame abstract elementary class categorical in a cardinal of sufficiently high cofinality admits a good frame: a forking-like notion for 1-types. This gives a construction of a good frame in ZFC from superstability and tameness, a locality property of types that (arguably) is likely to hold in practice. It follows for example that tameness and categoricity at a suitable cardinal imply stability everywhere. Subsequent works rely on this method and improve it further (for example the cofinality requirement is removed in "Independence in abstract elementary classes").
Will Boney and Sebastien Vasey, Tameness and frames revisited, The Journal of Symbolic Logic 82 (2017), no. 3, 995–1021. Publisher version pdf (see all versions) arXiv.
Will Boney has shown that assuming tameness for types of length two, a good λ-frame (a notion of forking for types of length one over models of size λ) transfers to a good frames for types of length one over models of size above λ. Tameness for types of length two, rather than length one, is used in Boney's proof of the symmetry property. In this paper, we replace tameness for types of length two by tameness for types of length one. The proof of symmetry is more conceptual and goes through studying independent sequences. We obtain as an interesting corollary of our methods that a well-behaved notion of dimension exists in this framework: any two maximal infinite independent sets have the same cardinality.
Sebastien Vasey, Infinitary stability theory, Archive for Mathematical Logic 55 (2016), nos. 3-4, 562–592. Publisher version pdf (see all versions) arXiv.
We introduce the Galois Morleyization of an AEC: a trick to think of semantic (Galois) types as being syntactic. This give a correspondence between AECs and the syntactic framework of stability theory inside a model. We use this to prove the equivalence between no order property and stability in tame AECs as well as a stability spectrum theorem in that context. We also improve results of Boney and Grossberg on the existence of a forking-like notion in stable, fully tame and short AECs. This gives evidence that some stability theory can be developped also in strictly stable AECs.
Sebastien Vasey, Building independence relations in abstract elementary classes, Annals of Pure and Applied Logic 167 (2016), no. 11, 1029–1092. Publisher version pdf (see all versions) arXiv.
We study general methods to build forking-like notions in the framework of tame AECs with amalgamation. We study here both local notions like good frames and global notions (for types of all lengths). We build a good frame from categoricity in a high-enough cardinal (no matter what its cofinality is), and give conditions under which the frame extends to a global independence notion. Modulo an unproven claim of Shelah, we deduce that Shelah's categoricity conjecture follows from the weak generalized continuum hypothesis and unboundedly many strongly compact cardinals.
Note 1: This used to be part of Infinitary stability theory but is now a separate paper.
Note 2: This used to be called "Independence in abstract elementary classes".
Will Boney and Sebastien Vasey, Chains of saturated models in AECs, Archive for Mathematical Logic 56 (2017), no. 3, 187–213. Publisher version pdf (see all versions) arXiv.
.We work in the framework of tame AECs with amalgamation satisfying a natural notion of superstability (a version of local character of nonsplitting). We show that above a certain Hanf number, unions of chains of λ-saturated models are λ-saturated, and limit models of size λ are unique. We deduce the final piece in the construction of a good frame from tameness and superstability and conclude that for a high-enough λ, a tame superstable AEC has a good frame with underlying class its λ-saturated models.
Another contribution of this paper is to develop the theory of averages for tame AECs. This uses the syntactic-semantic correspondence isolated in "Infinitary stability theory".
Sebastien Vasey, Shelah's eventual categoricity conjecture in universal classes: part I, Annals of Pure and Applied Logic 168 (2017), no. 9, 1609–1642. Publisher version pdf arXiv (see all versions).
Shelah's eventual categoricity conjecture for abstract elementary classes says that an AEC categorical in a high-enough cardinal is categorical on a tail of cardinals. It is widely recognized as the main test question in the field of classification theory for AECs. We prove an approximation assuming that the AEC is a universal class (roughly, the class of models of a universal sentence in infinitary logic): categoricity in cardinals of arbitrarily high cofinality implies categoricity on a tail. Moreover, assuming amalgamation, categoricity in a single cardinal above the second Hanf number H_{2} implies categoricity everywhere above H_{2}. This is all in ZFC without assuming that the categoricity cardinal is a successor. The method generalizes to AECs that satisfy a locality condition on types (full tameness and shortness) and have primes over models of the form Ma, where M is a model in the class and a is a singleton.
The proof goes by observing that in such classes, the existence of a good frame in a single cardinal implies amalgamation everywhere above the cardinal (this uses the upward frame transfer of Boney). Using the orthogonality calculus developped in Chapter III of Shelah's book on AECs as well as the existence of prime assumption, one can use the upward transfer once again on a suitable frame to prove that if the class is not categorical on a tail, it must have a non-saturated model in every cardinal. To obtain an initial good frame, we use deep results from Shelah's AEC book (Chapter IV).
Note 1: This used to be named "Shelah's eventual categoricity conjecture in universal classes", and before that "Amalgamation from categoricity in universal classes".
Note 2: Earlier versions claimed the full eventual categoricity conjecture (i.e. without the cofinality assumptions), but gaps were later discovered. The full result is proven in Part II.
Note 3: I have written an expository note giving a short outline of the main steps of the proof.
Rami Grossberg and Sebastien Vasey, Equivalent definitions of superstability in tame abstract elementary classes, The Journal of Symbolic Logic 82 (2017), no. 4, 1387–1408. Publisher version pdf (see all versions) arXiv.
Working in the framework of tame AECs with amalgamation, we show that many definitions of superstability are equivalent. This includes the Shelah-Villaveces definition in terms of locality of splitting, the uniqueness of limit models, the existence of a good frame, as well as solvability, a definition that Shelah hails as the true counterpart to first-order superstability in chapter IV of his book.
Note: This used to be named "Superstability in abstract elementary classes".
Monica VanDieren and Sebastien Vasey, Symmetry in abstract elementary classes with amalgamation, Archive for Mathematical Logic 56 (2017), no. 3, 423–452. Publisher version pdf (see all versions) arXiv.
We improve several results of Shelah's milestone paper #394 on categorical AECs with amalgamation, getting uniqueness of limit models from categoricity in a suitable limit cardinal. As a corollary, we deduce the existence of a good frame (a local notion of forking) in this framework and also show that the model in the categoricity cardinal has some saturation if the cardinal is big-enough (but potentially has low cofinality) . This furthers our understanding of AECs with amalgamation that are not necessarily tame. With tameness, we also improve work of Will Boney and the second author (from "Chains of saturated models in AECs") on getting uniqueness of limit models from superstability and tameness. For example, if a class is λ-superstable and λ-tame, then limit models of size λ are unique and right above λ, unions of chains of saturated models are saturated. These results are proven using the symmetry property for splitting, isolated by the first author. The main technicals tools are a downward transfer of symmetry and a way to obtain it from failure of the order property.
Note: The paper was previously named "Transferring symmetry downward and applications", but has since been merged with "On the structure of categorical abstract elementary classes with amalgamation".
Sebastien Vasey, Shelah's eventual categoricity conjecture in tame AECs with primes, Accepted (January 23, 2017), Mathematical Logic Quarterly. Preprint: pdf arXiv, 16 pages. Last updated on January 20, 2017 (see all versions).
We show that tame AECs with amalgamation which have primes over sets of the form Ma satisfy Shelah's eventual categoricity conjecture. This improves on the result in "Shelah's eventual categoricity conjecture in universal classes" which asked for the AEC to be fully tame and short, a stronger locality property than tameness. We also deduce Shelah's categoricity conjecture for homogeneous model theory.
Sebastien Vasey, Building prime models in fully good abstract elementary classes, Mathematical Logic Quarterly 63 (2017), nos. 3–4, 193–201. Publisher version pdf (see all versions) arXiv.
We prove the converse of a theorem in "Shelah's eventual categoricity conjecture in universal classes". The later paper showed that fully tame and short AECs with amalgamation and existence of primes over sets of the form Ma that are categorical in a high-enough cardinal and are categorical on a tail of cardinals. We show that a fully tame and short AEC with amalgamation categorical on a tail of cardinal has primes on a tail of cardinals. More generally, we show that, for any AEC with a superstable-like global independence notion its class of saturated models has primes. This generalizes an argument of Shelah who proved it for saturated models of successor size.
Note: The paper was previously called "On prime models in totally categorical abstract elementary classes".
Will Boney, Rami Grossberg, Michael Lieberman, Jiří Rosický, and Sebastien Vasey, μ-Abstract elementary classes and other generalizations, Journal of Pure and Applied Algebra 220 (2016), no. 9, 3048–3066. Publisher version pdf (see all versions) arXiv.
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.
Sebastien Vasey, Downward categoricity from a successor inside a good frame, Annals of Pure and Applied Logic 168 (2017), no. 3, 651–692. Publisher version pdf (see all versions) arXiv.
We prove a downward transfer from categoricity in a successor in tame AECs: If an AEC K is LS(K)-tame, has amalgamation, and no maximal models, and is categorical in a successor cardinal above H_{1}, then it is categorical in all cardinals above H_{1}. This complements the upward transfer of Grossberg and VanDieren and improves the Hanf number of H_{2} in Shelah's downward transfer (provided the class is tame). The argument uses orthogonality calculus and gives alternate proofs to both the Shelah and the Grossberg-VanDieren transfers. We deduce Shelah's categoricity conjecture (so the Hanf number is H_{1}) for universal classes with amalgamation. Heavily using results of Shelah and the weak generalized continuum hypothesis, we can also prove Shelah's categoricity conjecture for tame AECs with amalgamation.
Note: This was previously called "A downward categoricity transfer for tame abstract elementary classes".
Will Boney and Sebastien Vasey, A survey on tame abstract elementary classes, Beyond First Order Model Theory (José Iovino ed.), CRC Press (2017), 353–427. pdf (see all versions) arXiv.
We survey recent developments in the study of tame abstract elementary classes. We emphasize the role of abstract independence relations such as Shelah's good frames. As an application, we sketch a proof of a categoricity transfer in universal classes (due to the second author): If a universal class is categorical in cardinals of arbitrarily high cofinality, then it is categorical on a tail of cardinals.
Sebastien Vasey, Shelah's eventual categoricity conjecture in universal classes: part II, Selecta Mathematica 23 (2017), no. 2, 1469–1506. Publisher version pdf (see all versions) arXiv.
We prove Shelah's eventual categoricity conjecture in universal classes. The threshold for categoricity is below the second Hanf number. We show for example that a universal L_{ω1, ω} sentence that is categorical in some cardinal above B_{Bω1} (where B denotes the ב function) is categorical in all such cardinals.
We do not assume anything else: the categoricity cardinal need not be a successor, the class is not assumed to have amalgamation, and the proof is in ZFC (no large cardinals or weak GCH).
The proof heavily uses Shelah's structure theory for universal classes developped in Chapter V of his book on AECs (an earlier version is Shelah's paper #300), as well as the categoricity transfer for tame AECs with amalgamation that have primes over models of the form Ma.
Note: This continues Part I, which proved an approximation (in a more general framework) and developped tools that we use here.
Sebastien Vasey, Saturation and solvability in abstract elementary classes with amalgamation, Archive for Mathematical Logic 56 (2017), nos. 5-6, 671–690. Publisher version pdf (see all versions) arXiv.
We prove that in an AEC with amalgamation and no maximal models categorical above the Löwenheim-Skolem-Tarski number, the model in the categoricity cardinal is Galois-saturated. This answers a question asked independently by Baldwin and Shelah.
In the proof, we rely heavily on our work on the symmetry property with VanDieren. We also use results of Shelah on extracting a strictly indiscernible subsequence from any sequence and prove that categoricity implies failure of the order property, even of relatively small lengths.
We obtain several partial categoricity transfers, as well as other corollaries (for example on the uniqueness of limit models).
Will Boney and Sebastien Vasey, Good frames in the Hart-Shelah example, Accepted (November 7, 2017) . Archive for Mathematical Logic. Publisher version pdf (see all versions) arXiv, 26 pages. Last updated on November 8, 2017.
For a fixed natural number n≥1, the Hart-Shelah example is an abstract elementary class (AEC) with amalgamation that is categorical exactly in the infinite cardinals less than or equal to ℵ_{n}.
We investigate recently-isolated properties of AECs in the setting of this example. We isolate the exact amount of type-shortness holding in the example and show that it has a type-full good ℵ_{n−1}-frame which fails the existence property for uniqueness triples. This gives the first example of such a frame.
Sebastien Vasey, Toward a stability theory of tame abstract elementary classes, Submitted (October 26, 2016). Preprint: pdf arXiv, 34 pages. Last updated on February 15, 2018 (see all versions).
The paper initiates 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 mild cardinal arithmetic (e.g. SCH or working above a strongly compact), we give a full characterization of the eventual stability spectrum. We connect it to the uniqueness of limit models, the behavior of chains of saturated models, and the saturation spectrum. We conclude that if a class is stable on a tail of cardinals, then it is superstable (in any of the senses proven equivalent in the above paper with Grossberg). Thus there is a clear notion of superstability in the framework of tame AECs with amalgamation.
Will Boney, Rami Grossberg, Monica VanDieren, and Sebastien Vasey, Superstability from categoricity in abstract elementary classes, Annals of Pure and Applied Logic 168 (2017), no. 7, 1383–1395. Publisher version pdf (see all versions) arXiv.
Starting from an abstract elementary class with no maximal models, Shelah and Villaveces have shown (assuming instances of diamond) that categoricity implies a superstability-like property for a certain independence relation called nonsplitting. We generalize their result as follows: given an abstract notion of independence for Galois (orbital) types over models, we derive that the notion satisfies a superstability property provided that the class is categorical and satisfies a weakening of amalgamation. This extends the Shelah-Villaveces result (the independence notion there was splitting) as well as a result of the first and second author where the independence notion was coheir. The argument is in ZFC and fills a gap in the Shelah-Villaveces proof.
Sebastien Vasey, Quasiminimal abstract elementary classes, Accepted (June 19, 2017), Archive for Mathematical Logic. Publisher version pdf arXiv, 17 pages. Last updated on June 12, 2017 (see all versions).
We propose the notion of a quasiminimal abstract elementary class (AEC). This is an AEC satisfying four semantic conditions: countable Löwenheim-Skolem-Tarski number, existence of a prime model, closure under intersections, and uniqueness of nonalgebraic orbital types over every countable model. We exhibit a correspondence between Zilber's quasiminimal pregeometry classes and quasiminimal AECs: any quasiminimal pregeometry class induces a quasiminimal AEC (this was known), and for any quasiminimal AEC there is a natural functorial expansion that induces a quasiminimal pregeometry class. We show in particular that the exchange axiom is redundant in Zilber's definition of a quasiminimal pregeometry class.
Sebastien Vasey, On the uniqueness property of forking in abstract elementary classes, Mathematical Logic Quarterly 63 (2017), no. 6, 598–604. Publisher version pdf (see all versions) arXiv.
In the setup of abstract elementary classes satisfying a local version of superstability, we prove the uniqueness property for 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.
Michael Lieberman, Jiří Rosický, and Sebastien Vasey, Universal abstract elementary classes and locally multipresentable categories, Submitted (August 16, 2017) . Preprint: pdf arXiv, 14 pages. Last updated on August 16, 2017 (see all versions).
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.
Sebastien Vasey, Tameness from two successive good frames, Submitted (August 14, 2017) . Preprint: pdf arXiv, 25 pages. Last updated on August 14, 2017.
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 implies the existence of a superstable-like notion for models of cardinality λ^{+}, but here we prove the converse. An immediate consequence is that forking in λ^{+} can be described in terms of forking in λ.
Michael Lieberman, Jiří Rosický, and Sebastien Vasey, Internal sizes in μ-abstract elementary classes, Submitted (October 4, 2017) . Preprint: pdf arXiv, 26 pages. Last updated on October 4, 2017 (see all versions).
The internal size of an object M inside a given category is, roughly, the least infinite cardinal λ such that any morphism from M into the colimit of a λ^{+}-directed system factors through one of the components of the system. The existence spectrum of a category is the class of cardinals λ such that the category has an object of internal size λ. We study the existence spectrum in μ-abstract elementary classes (μ-AECs), which are, up to equivalence of categories, the same as accessible categories with all morphisms monomorphisms. We show for example that, assuming instances of the singular cardinal hypothesis which follow from a large cardinal axiom, μ-AECs which admit intersections have objects of all sufficiently large internal sizes. We also investigate the relationship between internal sizes and cardinalities and analyze a series of examples, including one of Shelah---a certain class of sufficiently-closed constructible models of set theory---which show that the categoricity spectrum can behave very differently depending on whether we look at categoricity in cardinalities or in internal sizes.
Saharon Shelah and Sebastien Vasey, Abstract elementary classes stable in ℵ_{0}, Submitted (February 27, 2017). Preprint: pdf arXiv, 27 pages. Last updated on February 1, 2018 (see all versions).
We study abstract elementary classes (AECs) that, in ℵ_{0}, have amalgamation, joint embedding, no maximal models and are stable (in terms of the number of orbital types). Assuming a locality property for types, we prove that such classes exhibit superstable-like behavior at ℵ_{0}. More precisely, there is a superlimit model of cardinality ℵ_{0} and the class generated by this superlimit has a type-full good ℵ_{0}-frame (a local notion of nonforking independence) and a superlimit model of cardinality ℵ_{1}. We also give a supersimplicity condition under which the locality hypothesis follows from the rest.
Marcos Mazari Armida and Sebastien Vasey, Universal classes near ℵ_{1}, Submitted (December 31, 2017). Preprint: pdf arXiv, 12 pages. Last updated on December 31, 2017 (see all versions).
Shelah has provided sufficient conditions for an L_{ω1, ω}-sentence ψ to have arbitrarily large models and for a Morley-like theorem to hold of ψ. These conditions involve structural and set-theoretic assumptions on all the ℵ_{n}'s. Using tools of Boney, Shelah, and the second author, we give assumptions on ℵ_{0} and ℵ_{1} which suffice when ψ is restricted to be universal:
Assume 2^{ℵ0} < 2^{ℵ1}. Let ψ be a universal L_{ω1, ω}-sentence.
The theorem generalizes to the framework of L_{ω1, ω}-definable tame abstract elementary classes with primes.
Will Boney and Sebastien Vasey, Structural logic and abstract elementary classes with intersections, Submitted (January 16, 2018) . Preprint: pdf arXiv, 14 pages. Last updated on January 16, 2018 (see all versions).
We give a syntactic characterization of abstract elementary classes (AECs) closed under intersections using a new logic with a quantifier for isomorphism types that we call structural logic: we prove that AECs with intersections correspond to classes of models of a universal theory in structural logic. This generalizes Tarski's syntactic characterization of universal classes.
Michael Lieberman, Jiří Rosický, and Sebastien Vasey, Forking independence from the categorical point of view, Submitted (February 7, 2018) . Preprint: pdf arXiv, 48 pages. Last updated on February 7, 2018 (see all versions).
We develop the theory of forking-like independence in accessible categories. To do so, we present an axiomatic definition of what we call a stable independence notion and show that this is in fact a purely category-theoretic axiomatization of the properties of model-theoretic forking in a stable first-order theory.
We then show that any coregular locally presentable category with effective unions admits a forking-like independence notion. This includes the cases of Grothendieck toposes and Grothendieck abelian categories.
We also give conditions for existence and canonicity of stable independence notions. Specifically, an accessible category with directed colimits whose morphisms are monomorphisms will have at most one stable independence notion. Moreover, assuming a large cardinal axiom, a cofinal full subcategory will have a stable independence notion if and only if a certain order property fails. This establishes, in particular, a category-theoretic characterization of stability (in the model-theoretic sense) for accessible categories.
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