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 H2 implies categoricity everywhere above H2. 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, Mathematical Logic Quarterly 64 (2018), nos. 1–2, 25–36. Publisher version pdf (see all versions) arXiv.
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 H1, then it is categorical in all cardinals above H1. This complements the upward transfer of Grossberg and VanDieren and improves the Hanf number of H2 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 H1) 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 Lω1, ω sentence that is categorical in some cardinal above BBω1 (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, Archive for Mathematical Logic 57 (2018), nos. 5-6, 687–712. Publisher version pdf (see all versions) arXiv.
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 ℵn.
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 ℵn−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, Journal of Mathematical Logic 18 (2018), no. 2, 1850009 (34 pages). Publisher version pdf (see all versions) arXiv.
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, Archive for Mathematical Logic 57 (2018), nos. 3-4, 299–315. Publisher version pdf (see all versions) arXiv.
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, Proceedings of the American Mathematical Society 147 (2019), no. 3, 1283–1298. Publisher version pdf (see all versions) arXiv.
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, Israel Journal of Mathematics 235 (2020), 465–500. Publisher version pdf arXiv.
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, Journal of Pure and Applied Algebra 223 (2019), no. 10, 4560–4582. Publisher version pdf (all versions) arXiv.
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 ℵ0, Annals of Pure and Applied Logic 169 (2018), no. 7, 565–587. Publisher version pdf (see all versions) arXiv.
We study abstract elementary classes (AECs) that, in ℵ0, 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 ℵ0. More precisely, there is a superlimit model of cardinality ℵ0 and the class generated by this superlimit has a type-full good ℵ0-frame (a local notion of nonforking independence) and a superlimit model of cardinality ℵ1. We also give a supersimplicity condition under which the locality hypothesis follows from the rest.
Marcos Mazari-Armida and Sebastien Vasey, Universal classes near ℵ1, The Journal of Symbolic Logic 83 (2018), no. 4, 1633–1643. Publisher version pdf (all versions) arXiv.
Shelah has provided sufficient conditions for an Lω1, 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 ℵn's. Using tools of Boney, Shelah, and the second author, we give assumptions on ℵ0 and ℵ1 which suffice when ψ is restricted to be universal:
Assume 2ℵ0 < 2ℵ1. Let ψ be a universal Lω1, ω-sentence.
The theorem generalizes to the framework of Lω1, ω-definable tame abstract elementary classes with primes.
Will Boney and Sebastien Vasey, Structural logic and abstract elementary classes with intersections, Bulletin of the Polish Academy of Sciences (Mathematics) 67 (2019), 1–17. Publisher version pdf (all versions) arXiv.
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, Advances in Mathematics 346 (2019), 719–772. Publisher version pdf (all versions) arXiv.
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 category 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.
Nathanael Ackerman, Will Boney, and Sebastien Vasey, Categoricity in multiuniversal classes, Annals of Pure and Applied Logic 170 (2019), no. 11, 102712 (15 pages). Publisher version pdf (all versions) arXiv.
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.
Sebastien Vasey, The categoricity spectrum of large abstract elementary classes, Selecta Mathematica 23 (2019), no. 5, 65 (51 pages). Publisher version pdf (see all versions) arXiv.
The categoricity spectrum of a class of structures is the collection of cardinals in which the class has a single model up to isomorphism. Assuming that cardinal exponentiation is injective (a weakening of the generalized continuum hypothesis, GCH), we give a complete list of the possible categoricity spectrums of an abstract elementary class with amalgamation and arbitrarily large models. Specifically, the categoricity spectrum is either empty, an end segment starting below the Hanf number, or a closed interval consisting of finite successors of the Löwenheim-Skolem-Tarski number (there are examples of each type). We also prove (assuming a strengthening of the GCH) that the categoricity spectrum of an abstract elementary class with no maximal models is either bounded or contains an end segment. This answers several longstanding questions around Shelah's categoricity conjecture.
Saharon Shelah and Sebastien Vasey, Categoricity and multidimensional diagrams, Submitted (May 23, 2018). Preprint: pdf (see all versions) arXiv, 63 pages. Last updated on May 23, 2018.
We study multidimensional diagrams in independent amalgamation in the framework of abstract elementary classes (AECs). We use them to prove the eventual categoricity conjecture for AECs, assuming a large cardinal axiom. More precisely, we show assuming the existence of a proper class of strongly compact cardinals that an AEC which has a single model of some high-enough cardinality will have a single model in any high-enough cardinal. Assuming a weak version of the generalized continuum hypothesis, we also establish the eventual categoricity conjecture for AECs with amalgamation.
Sebastien Vasey, On categoricity in successive cardinals, to appear in The Journal of Symbolic Logic. Preprint: pdf (see all versions) arXiv, 19 pages. Last updated on March 19, 2020.
We investigate, in ZFC, the behavior of abstract elementary classes (AECs) categorical in many successive small cardinals. We prove for example that a universal Lω1, ω sentence categorical on an end segment of cardinals below the first uncountable strong limit must be categorical also everywhere above the first uncountable strong limit. 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.
Michael Lieberman, Jiří Rosický, and Sebastien Vasey, Sizes and filtrations in accessible categories, to appear (accepted June 17, 2019) in the Israel Journal of Mathematics. Publisher version pdf (all versions) arXiv, 36 pages. Last updated on June 5, 2019.
Note: This was previously named "set-theoretic aspects of accessible categories".
An accessible category is, roughly, a category with all sufficiently directed colimits, in which every object can be resolved as a directed system of "small" subobjects. Such categories admit a purely category-theoretic replacement for cardinality: the internal size. Generalizing results and methods from "Internal sizes in μ-abstract elementary classes", we examine set-theoretic problems related to internal sizes and prove several Löwenheim-Skolem theorems for accessible categories. For example, assuming the singular cardinal hypothesis, we show that a large accessible category has an object in all internal sizes of high-enough cofinality. We also introduce the notion of a filtrable accessible category - one in which any object can be represented as the colimit of a chain of strictly smaller objects - and examine the conditions under which an accessible category is filtrable. We prove (modulo a small technical condition) that accessible categories with directed colimit are filtrable.
Michael Lieberman, Jiří Rosický, and Sebastien Vasey, Cellular categories and stable independence, Submitted (May 1, 2019). Preprint: pdf (all versions) arXiv, 22 pages. Last updated on March 3, 2020.
Note: This was previously named "weak factorization systems and stable independence".
We exhibit a bridge between the theory of cellular categories, used in algebraic topology and homological algebra, and the model-theoretic notion of stable independence. Roughly speaking, we show that the combinatorial cellular categories (those where, in a precise sense, the cellular morphisms are generated by a set) are exactly those that give rise to stable independence notions. We give two applications: on the one hand, we show that the abstract elementary classes of roots of Ext studied by Baldwin-Eklof-Trlifaj are stable and tame. On the other hand, we give a simpler proof (in a special case) that combinatorial categories are closed under 2-limits, a theorem of Makkai and Rosický.
Sebastien Vasey, Accessible categories, set theory, and model theory: an invitation, Submitted (April 25, 2019). Preprint: pdf (all versions) arXiv, 68 pages. Last updated on January 7, 2020.
We give a self-contained introduction to accessible categories and how they shed light on both model- and set-theoretic questions. We survey for example recent developments on the study of presentability ranks, a notion of cardinality localized to a given category, as well as stable independence, a generalization of pushouts and model-theoretic forking that may interest mathematicians at large. We give many examples, including recently discovered connections with homotopy theory and homological algebra. We also discuss concrete versions of accessible categories (such as abstract elementary classes), and how they allow nontrivial "element by element" constructions. We conclude with a new proof of the equivalence between saturated and homogeneous which does not use the coherence axiom of abstract elementary classes.
Michael Lieberman, Jiří Rosický, and Sebastien Vasey, Hilbert spaces and C*-algebras are not finitely concrete, Submitted (September 16, 2019). Preprint: pdf arXiv, 7 pages. Last updated on September 20, 2019.
We show that no faithful functor from the category of Hilbert spaces with linear isometries into the category of sets preserves directed colimits. Thus Hilbert spaces cannot form an abstract elementary class, even up to change of language. We deduce an analogous result for the category of commutative unital C*-algebras with *-homomorphisms. This implies, in particular, that this category is not axiomatizable by a first-order theory, a strengthening of a conjecture of Bankston.
Michael Lieberman, Leonid Positselski, Jiří Rosický, and Sebastien Vasey, Cofibrant generation of pure monomorphisms, Submitted (January 15, 2020). Preprint: pdf arXiv, 14 pages. Last updated on January 11, 2020.
We show that pure monomorphisms are cofibrantly generated (generated from a set of morphisms by pushouts, transfinite composition, and retracts) in any locally finitely presentable additive category. In particular, this is true in any category of R-modules.
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