Toward a Categoricity Theorem for Abstract Elementary Classes

Monica VanDieren

Carnegie Mellon University

www.math.cmu.edu/~monicav

Rutgers University 2001 Logic Conference: MAMLS

October 13, 2001

Non-Elementary Classes

There are many natural mathematical classes that cannot be captured by first order logic:

1. Archimedian ordered fields

2. Locally finite groups

3. Well-ordered sets

4. Noetherian Rings, etc.

Extensions of First-order Logic

1. L_{κ,ω} when κ > ω

2. L(Q) when Q is interpretted as the Keisler-quantifier (there exist at least ℵ_{1} many)

3. L_{ω_{1},ω}(Q)

Definition. Let L_{1} be an expansion of the language L. Suppose T_{1} is a first order theory in L_{1} and Γ is a collection of first order types without parameters in the language of L_{1}. We denote by

P C(T_{1}, Γ, L) :=

{(M ↾ L) : M ⊧ T_{1} and M omits every type from Γ}.

If K is a class of models such that K = P C(T_{1}, Γ, L(K)) for some T_{1}, Γ and L_{1} with μ = |T_{1}|+|Γ|+|L_{1}|+ℵ_{0}, then we call K a P C_{μ}-class.

Abstract elementary classes

Definition. Let K be a class of structures all in the same similarity type L(K), and let ≺_{K} be a partial order on K. The ordered pair ⟨K, ≺_{K}⟩ is an abstract elementary class iff

A0 (Closure under isomorphism)

(a) For every M ∈ K and every L(K)-structure N if M ≅ N then N ∈ K.

(b) Let N_{1}, N_{2} ∈ K and M_{1}, M_{2} ∈ K such that there exist f_{l} : N_{l} ≅ M_{l} (for l = 1, 2) satisfying f_{1} ⊆ f_{2} then N_{1} ≺_{K} N_{2} implies that M_{1} ≺_{K} M_{2}.

A1 For all M, N ∈ K if M ≺_{K} N then M ⊆ N.

A2 Let M, N, M^{∗} be L(K)-structures. If M ⊆ N, M ≺_{K} M^{∗} and N ≺_{K} M^{∗} then M ≺_{K} N.

A3 (Löwenheim-Skolem) There exists a cardinal LS(K) ≥ +ℵ_{0} + |L(K)| such that for every M ∈ K and for every A ⊆ |M| there exists N ∈ K such that N ≺_{K} M, |N| ⊇ A and ∥N∥ ≤ |A| + LS(K).

A4 (Tarski-Vaught Chain)

(a) For every regular cardinal μ and every N ∈ K if {M_{i} ≺_{K} N : i < μ} ⊆ K is increasing (i.e. i < j ⟹ M_{i} ≺_{K} M_{j}) then ⋃_{i<μ} M_{i} ∈ K and ⋃_{i<μ} M_{i} ≺_{K} N.

(b) For every regular μ, if {M_{i} : i < μ} ⊆ K is increasing then ⋃_{i<μ} M_{i} ∈ K and M_{0} ≺_{K} ⋃_{i<μ} M_{i}.

Shelah’s Presentation Theorem. If K is an AEC, then K is a P C_{μ}-class for some μ ≤ 2^{L(K)}.

Notation.

1. For M, N ∈ K a monomorphism f : M → N is called a ≺_{K}-embedding iff f[M] ≺_{K} N. Denote this by f : M ↪_{K} N.

2. K_{λ} := {M ∈ K : ∥M∥ = λ}.

3. K_{<λ} := {M ∈ K : ∥M∥ < λ}.

In pure model theory, through solutions to �ǒs Conjecture, central concepts for the classification of first order theories were identified (ie. prime models, minimal types, super-stability, etc.).

�ǒs conjecture.(1956) Let T be a first-order theory. If there exists λ > |T | + ℵ_{0} such that I(λ, T) = 1 then I(μ, T) = 1 holds for every μ > |T | + ℵ_{0}.

Our goal in studying AECs is to develop a classification theory for non-elementary classes. A test question for the proposed stability theory was identified by Shelah in the 70s for L_{ω_{1},ω}. Later this conjecture was generalized to a categoricity transfer theorem for AECs.

Shelah’s conjecture.(about 1977) Let K be an AEC. If K is categorical in some λ ≥ Hanf(K), then K is categorical in every μ ≥ Hanf(K).

Status - very open. Partial results:

1. Keisler (1971), for L_{ω_{1},ω} under the additional assumption of existence sequentially homogeneous model.

2. Shelah (1984) [87a] and [87b] for excellent classes.

3. Lessmann (1998) for countable finite diagrams.

4. Makkai and Shelah [Sh285] for L_{κ,ω}-theories under the additional assumption that κ is a strongly compact cardinal, AND λ = χ^{+} .

5. Kolman-Shelah [Sh362] and Shelah [Sh 472] partial going down results for L_{κ,ω}-theories where κ is a measurable cardinal and λ = χ^{+} .

6. Shelah [Sh394] partial going down result for AECs with the Amalgamation Property under the assumption that λ = χ^{+} .

Context of Shelah and Villaveces’ Work

1. GCH holds,

2. K is an abstract elementary class,

3. K has no maximal models and

4. K is categorical in some λ with λ ≥ LS(K).

The notion of Galois types

Definition. K_{μ}^{3} := {(M, N, a) | a ∈ N; M ≺_{K} N; M, N ∈ K_{μ}}. Definition. Let M ∈ K_{μ}, We say that (M, N_{1}, a_{1}) and (M, N_{2}, a_{2}) ∈ K_{μ}^{3}, are ∼-related, written

(M, N_{1}, a_{1}) ∼ (M, N_{2}, a_{2}),

if there exists N ∈ K_{λ} and ≺_{K}-embeddings

h_{1} : N_{1} → N and h_{2} : N_{2} → N, such that

h_{1}(a_{1}) = h_{2}(a_{2})

and the following diagram commutes:

\xymatrix{N_{1} \ar[r]^{h_{1}} & N \\ M \ar[u]^{id} \ar[r]_{id} & N_{2} \ar[u]_{h_{2}}}

Amalgamation Property

Definition. Let (K, ≺_{K}) be an abstract elementary class, and let μ ≥ LS(K) be a cardinal.

1. We say that M_{0} ∈ K_{μ} is an amalgamation base iff for every M_{1}, M_{2} ∈ K_{μ} and every f_{ℓ} : M_{0} ↪_{K} M_{ℓ} for ℓ = 1, 2, there exists N ∈ K_{μ} so that there are g_{1} : M_{1} ↪_{K} N and g_{2} : M_{2} ↪_{K} N so that g_{1} ∘ f_{1} ↾ |M_{0}| = g_{2} ∘ f_{2} ↾ |M_{0}|. Namely, the following diagram is commutative:

\xymatrix{M_{1} \ar[r]^{g_{1}} & N \\ M_{0} \ar[u]^{f_{1}} \ar[r]_{f_{2}} & M_{2} \ar[u]_{g_{2}}}

2. Denote by K_{μ}^{am} the collection of models in K_{μ} that are amalgamation bases.

Definition. We say that K satisfies the amalgamation property (AP) iff every model M ∈ K is an amalgamation base.

Types

Proposition. ∼ is an equivalence relation on {(M, N, ā) ∈ K_{μ}^{3} | M, N are amalgamation bases }.

Definition

1. For M ≺_{K} N ∈ K_{μ}^{am} and a ∈ N, we let

tp(a/M, N) := (M, N, a)/ ∼ . This is the type of a over M in N.

2. For M ∈ K_{μ}^{am}, we let S(M) := { tp(a′/M, N) : M ≺_{K} N ∈ K_{μ}^{am}, a ∈ N}.

Density of Amalgamation Bases in the context of Shelah and Villaveces

Theorem. [Shelah-Villaveces] Let μ be a cardinal with LS(K) ≤ μ < λ. Then for every M ∈ K_{μ}, there exists M′ ∈ K_{μ}^{am} with M ≺_{K} M′.

Theorem. [V] Let μ be a cardinal with LS(K) ≤ μ < λ. Fix N_{λ} ∈ K_{λ} Suppose M ∈ K_{μ} and ā ∈ ^{μ}| N_{λ}|. If M ≺_{K} N_{λ}, then there exists Mā an amalgamation base of cardinality μ such that M ⋃ ā ⊆ Mā ≺_{K} N_{λ}.

Saturated Models

Definition. A model N ∈ K_{μ} is saturated iff for every M ≺_{K} N with M ∈ K_{<μ}^{am} and every type p ∈ S(M), there exists a realization of p in N.

Definition. An AEC, K, is μ-stable iff for every M ∈ K_{μ}^{am}, we have that |S(M)| ≤ μ.

Proposition. If K is μ-stable and has the amalgamation property, then there exists a saturated model N of cardinality μ. Moreover, the saturated model of cardinality μ is unique up to isomorphism.

Remark. Without the amalgamation property, the existence of a saturated model is not known in general.

Rough Sketch of Attempts towards the Categoricity Conjecture

Show that:

1. For every μ there exists a saturated model of cardinality μ.

2. Limit models of a given cardinality are unique up to isomorphism.

3. A model is saturated iff it is a limit model.

4. If M ∈ K_{μ} is not saturated, then there exists N ∈ K_{λ} which is not saturated.

Limit Models

Definition. We say N ∈ K is universal over M iff for every M′ ∈ K_{|M|} with M ≺_{K} M′, there exists f : M′ ↪ N such that f ↾ M = id_{M}.

Definition. Let μ be a cardinal and θ a limit ordinal with θ < μ^{+} . We say that N is a (μ, θ)-limit over M provided there exists ⟨N_{i} | i < θ⟩ such that for every i < θ,

1. N_{i} ∈ K_{μ}^{am}

2. N_{i+1} is universal over N_{i}

3. N_{0} = M and

4. ⋃_{i<θ} N_{i} = N.

Existence of Limit Models

Theorem. [Shelah-Villaveces] Let μ < λ. For every M ∈ K_{μ}^{am}, there exists N ∈ K_{∥M∥} with N universal over M.

Corollary. [Shelah-Villaveces] Let μ be a cardinal and θ a limit ordinal such that θ < μ^{+} ≤ λ. Then there exists a (μ, θ)-limit, M ∈ K.

Theorem. [Shelah-Villaveces] If M is a (μ, θ)-limit for some θ < μ^{+} ≤ λ, then M is an amalgamation base.

Theorem. [V] Let μ < λ. For every M ∈ K_{μ}^{am}, there exists N ∈ K_{μ^{+}} such that N is μ^{+} -universal over M.

Definition. We say N ∈ K is κ-universal over M ∈ K_{μ} iff for every M′ ∈ K_{κ} with M ≺_{K} M′, there exists f : M′ ↪ N with f ↾ M = id_{M}.

Uniqueness of Limit Models

Let μ < λ and θ a limit ordinal with θ < μ^{+} .

Conjecture. Let μ < λ and θ_{1}, θ_{2} be limit ordinals with θ_{1}, θ_{2} < μ^{+} . If N_{1} and N_{2} are (μ, θ_{1})- and (μ, θ_{2})-limits over M, respectively, then N_{1} ≅ N_{2}.

Proposition. [Shelah-Villaveces] If N_{1} and N_{2} are (μ, θ)-limits over M, then there exists f : N_{1} ≅ N_{2} with f ↾ M = id_{M}.

Proposition. [Shelah-Villaveces] If N_{1} is a (μ, θ)- limit over M and N_{2} is a (μ, cf (θ))-limit over M, then N_{1} ≅ N_{2}.

Towards Uniqueness of Limit Models

Approach the Uniqueness Conjecture in stages:

1. Prove that if N ∈ K_{λ} is weakly model homogeneous, then any 2 limit models of the same cardinality are isomorphic.

2. Prove that N ∈ K_{λ} is weakly model homogeneous.

Model Homogeneity

Definition. N is model homogeneous iff for every M ≺_{K} N with M ∈ K_{<∥N∥}, we have that N is universal over M.

Remark. If N ∈ K_{λ} (where λ is the categoricity cardinal) is model homogeneous, then K has the amalgamation property below λ.

Proposition. [Shelah] If K satisfies the amalgamation property , M ∈ K is saturated iff M is model homogeneous.

Remark. In particular, if K has the amalgamation property and is categorical in λ, then N ∈ K_{λ} is saturated and hence model homogeneous.

Definition. N is weakly model homogeneous (wmh) iff for every M ≺_{K} N with M ∈ K_{<∥N∥}^{am}, we have that N is universal over M.

In the context of Shelah-Villaveces assuming that N ∈ K_{λ} weakly model homogeneous:

1. [easy] N ∈ K_{λ} is saturated.

2. [V] If M is a (μ, θ)-limit for some θ < μ^{+} ≤ λ, then there exists an increasing and continuous sequence of ordinals ⟨α_{i} | i < θ⟩ such that

M ≅ EM(I_{sup_{i<θ}{α_{i}}}, Φ) ↾ L(K)

and ⟨EM(I_{α_{i}}, Φ) ↾ L(K) | i < θ⟩ witnesses that EM(I_{sup_{i<θ}{α_{i}}}, Φ) ↾ L(K) is a (μ, θ)-limit.

3. [V] (Uniqueness of limit models) Suppose θ_{l} is a limit ordinal with θ_{l} < μ^{+} ≤ λ (for l = 1, 2). If N_{l} are (μ, θ_{l})-limits for l = 1, 2, then there exists an isomorphism f : N_{1} ≅ N_{2}.

Conjecture. In the context of Shelah and Villaveces, the categoricity model is weakly model homogeneous.

Definition. Θ_{μ^{+}} is said to hold if and only if for all {f_{η} | η ∈ ^{μ^{+}} 2 where f_{η} : μ^{+} → μ^{+} } and for every club C ⊆ μ^{+} , there exists η ≠ ν ∈ ^{μ^{+}} 2 and there exists a δ ∈ C such that

1. η ↾ δ = ν ↾ δ,

2. f_{η} ↾ δ = f_{ν} ↾ δ and

3. η[δ] ≠ ν[δ]. Remark Devlin and Shelah have shown that Θ_{μ^{+}} follows from 2^{μ} < 2^{μ^{+}} .