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Shelah has provided sufficient conditions for an ${\Bbb L}_{\omega _1 ,\omega } $ -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 ${\aleph _n}$ 's. Using tools of Boney, Shelah, and the second author, we give assumptions on ${\aleph _0}$ and ${\aleph _1}$ which suffice when ψ is restricted to be universal: Theorem. Assume ${2^{{\aleph _0}}}. Let ψ be a universal ${\Bbb L}_{\omega _1 ,\omega } $ -sentence. (1) If ψ is categorical in ${\aleph _0}$ and $1 \leqslant {\Bbb L}\left({\psi ,\aleph _1 } \right) , then ψ has arbitrarily large models and categoricity of ψ in some uncountable cardinal implies categoricity of ψ in all uncountable cardinals. (2) If ψ is categorical in ${\aleph _1}$ , then ψ is categorical in all uncountable cardinals. The theorem generalizes to the framework of ${\Bbb L}_{\omega _1 ,\omega } $ -definable tame abstract elementary classes with primes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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:Corollary.LetKbe a tame AEC with a monster model. Assume thatKis stable in a proper class of cardinals. The following are equivalent:(1)For all high-enough λ,Khas no long splitting chains.(2)For all high-enough λ, there exists a good λ-frame on a skeleton ofKλ.(3)For all high-enough λ,Khas a unique limit model of cardinality λ.(4)For all high-enough λ,Khas a superlimit model of cardinality λ.(5)For all high-enough λ, the union of any increasing chain of λ-saturated models is λ-saturated.(6)There exists μ such that for all high-enough λ,Kis (λ,μ) -solvable.This gives evidence that there is a clear notion of superstability in the framework of tame AECs with a monster model. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

We study when a union of saturated models is saturated in the framework of tame abstract elementary classes (AECs) with amalgamation. We prove: Theorem 0.1. If K is a tame AEC with amalgamation satisfying a natural definition of superstability (which follows from categoricity in a high-enough cardinal), then for all high-enough $$\lambda {:}$$ Our proofs use independence calculus and a generalization of averages to this non first-order context. [ABSTRACT FROM AUTHOR]

In the setting of abstract elementary classes (AECs) with amalgamation, Shelah has proven a downward categoricity transfer from categoricity in a successor and Grossberg and VanDieren have established an upward transfer assuming in addition a locality property for Galois types that they called tameness. We further investigate categoricity transfers in tame AECs. We use orthogonality calculus to prove a downward transfer from categoricity in a successor in AECs that have a good frame (a forking-like notion for types of singletons) on an interval of cardinals: Theorem 0.1 Let K be an AEC and let LS ( K ) ≤ λ < θ be cardinals. If K has a type-full good [ λ , θ ] -frame and K is categorical in both λ and θ + , then K is categorical in all μ ∈ [ λ , θ ] . We deduce improvements on the threshold of several categoricity transfers that do not mention frames. For example, the threshold in Shelah's transfer can be improved from ℶ ℶ ( 2 LS ( K ) ) + to ℶ ( 2 LS ( K ) ) + assuming that the AEC is LS ( K ) -tame. The successor hypothesis can also be removed from Shelah's result by assuming in addition either that the AEC has primes over sets of the form M ∪ { a } or (using an unpublished claim of Shelah) that the weak generalized continuum hypothesis holds. [ABSTRACT FROM AUTHOR]

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. Theorem 0.1 Superstability from categoricity Let K be a ( < κ ) -tame AEC with amalgamation. If κ = ℶ κ > LS ( K ) and K is categorical in a λ > κ , then: • K is stable in any cardinal μ with μ ≥ κ . • K is categorical in κ. • There is a type-full good λ-frame with underlying class K λ . Under more locality conditions, we prove that the frame extends to a global independence notion (for types of arbitrary length). Theorem 0.2 A global independence notion from categoricity Let K be a densely type-local, fully tame and type short AEC with amalgamation. If K is categorical in unboundedly many cardinals, then there exists λ ≥ LS ( K ) such that K ≥ λ admits a global independence relation with the properties of forking in a superstable first-order theory. As an application, we deduce (modulo an unproven claim of Shelah) that Shelah's eventual categoricity conjecture for AECs (without assuming categoricity in a successor cardinal) follows from the weak generalized continuum hypothesis and a large cardinal axiom. Corollary 0.3 Assume 2 λ < 2 λ + for all cardinals λ, as well as an unpublished claim of Shelah. If there exists a proper class of strongly compact cardinals, then any AEC categorical in some high-enough cardinal is categorical in all high-enough cardinals. [ABSTRACT FROM AUTHOR]

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 nonsplitting, a particular notion of independence. We generalize their result as follows: given any 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. [ABSTRACT FROM AUTHOR]

We introduce μ -Abstract Elementary Classes ( μ -AECs) as a broad framework for model theory that includes complete boolean algebras and metric spaces, 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: for example, the preliminary classification-theoretic results for μ -AECs transfer directly to accessible categories with monomorphisms. [ABSTRACT FROM AUTHOR]