A topology for galois types in abstract elementary classes

Lieberman, Michael

doi:10.1002/malq.200910132

Cited by 5 publications

(3 citation statements)

References 14 publications

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“…While the standard treatment in AECs considers the set of types over a model M as a discrete set-or possibly a topological space, as in [15]-this is not in the spirit of mAECs. Associating types with orbits of elements in C, which is itself a metric structure, we obtain a pseudometric on the set of types over M: let d be the infimum of the distances between elements of respective orbits.…”

Section: Stabilitymentioning

confidence: 99%

Metric Abstract Elementary Classes as Accessible Categories

Lieberman¹,

Rosický²

2017

J. symb. log.

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We show that metric abstract elementary classes (mAECs) are, in the sense of [16], coherent accessible categories with directed colimits, with concrete ℵ 1directed colimits and concrete monomorphisms. More broadly, we define a notion of κ-concrete AEC-an AEC-like category in which only the κ-directed colimits need be concrete-and develop the theory of such categories, beginning with a categorytheoretic 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 [16] yield a proof that any categorical mAEC is µ-d-stable in many cardinals below the categoricity cardinal. 2 M. LIEBERMAN AND J. ROSICKÝ(4) (K, U) is a coherent accessible category with concrete directed colimits and concrete monomorphisms, where "coherence" is a property of U corresponding to the coherence axiom for AECs. (5) (K, U) is a coherent accessible category with concrete directed colimits and concrete monomorphisms, and satisfies the iso-fullness condition described in Remark 3.5 in [16]-such a category is equivalent to an AEC.Certain essential results from the theory of AECs are shown to hold at greater levels of generality: categories of the form (2) satisfy a presentation theorem generalizing that of Shelah and, if large, admit a robust EM-functor. Categories of the form (3) allow the development of Galois types and satisfy a generalization of Boney's theorem on tameness under the assumption of a proper class of strongly compact cardinals (see [7]). Categories of the form (4) satisfy the essential technical condition that Galois saturation corresponds to, in AEC terms, model-homogeneity, and support the development of a fragment of classification theory. We show in Section 3 that any mAEC K is an accessible category with directed colimits, and note that, if we take U : K → Set to be the usual underlying set functor, K is coherent with concrete monomorphisms. As is well known, though, directed colimits in K need not be concrete: when taking the colimit of a chain of structures in K, we must, in general, take the completion of the union of the underlying sets. This would seem to place us, at best, in type (2) above, which is already sufficient to give a presentation theorem and guarantee the existence of an EM-functor for a general mAEC-this is in itself a generalization of [10], the results of which hold only in the homogenous case. As we note in Remark 2.9, however, mAECs do have concrete ℵ 1 -directed colimits, which suggests that we may benefit from a generalization of the hierarchy of [16], considering categories with concrete κ-directed colimits for some κ. In case a category of this form has (not necessarily concrete) directed colimits, is coherent, has concrete monomorphisms, and is suitably replete and iso-full-the conditions of (4) above-we call it a κ-concrete AEC, or κ-CAEC for short. Incidentally, there is an alternative option already being pursued in, e.g., [8], namely to consider classes of structu...

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Section: Stabilitymentioning

confidence: 99%

Metric Abstract Elementary Classes as Accessible Categories

Lieberman¹,

Rosický²

2017

J. symb. log.

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“…Many results only use the assumption of tameness (e.g., ), while others use full tameness and shortness (but it is also unclear whether it is really needed there, cf. [, Question 15.4]).…”

Section: Introductionmentioning

confidence: 99%

Shelah's eventual categoricity conjecture in tame abstract elementary classes with primes

Vasey

2018

Mathematical Logic Qtrly

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A new case of Shelah's eventual categoricity conjecture is established: Let scriptK be an abstract elementary class with amalgamation. Write μ:=ℶfalse(2 LS false(scriptKfalse)false)+ and H2:=ℶfalse(2μfalse)+. Assume that scriptK is H2‐tame and K≥H2 has primes over sets of the form M∪{a}. If scriptK is categorical in some λ>H2, then scriptK is categorical in all λ′≥H2. The result had previously been established when the stronger locality assumptions of full tameness and shortness are also required. An application of the method of proof of the mentioned result is that Shelah's categoricity conjecture holds in the context of homogeneous model theory (this was known, but our proof gives new cases): If D be a homogeneous diagram in a first‐order theory T and D is categorical in a λ>|T|, then D is categorical in all λ′≥minfalse(λ,ℶ(2|T|)+false).

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“…Examples of the use of tameness and amalgamation include [BKV06] (an upward stability transfer), [Lie11] (showing that tameness is equivalent to a natural topology on Galois types being Hausdorff), [GV06c] (an upward categoricity transfer theorem, which can be combined with Fact 1.1 and the downward transfer of Shelah [She99] to prove that Shelah's eventual categoricity conjecture for a successor follows from the existence of a proper class of strongly compact cardinals) and [Bon14a,BVc,Jar16], showing that good frames behave well in tame classes.…”

Section: Introductionmentioning

confidence: 99%

Building independence relations in abstract elementary classes

Vasey

2016

Annals of Pure and Applied Logic

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Abstract. 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 = 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.

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A topology for galois types in abstract elementary classes

Cited by 5 publications

References 14 publications

Metric Abstract Elementary Classes as Accessible Categories

Metric Abstract Elementary Classes as Accessible Categories

Shelah's eventual categoricity conjecture in tame abstract elementary classes with primes

Building independence relations in abstract elementary classes

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