20 Years of Tameness

Monica VanDieren

November 17, 2021

Tameness

Limit Models

First Order Theories

\(L_{\omega_1,\omega}\) Theories

Incomplete Map of Non-elementary Classes

First Order Theories

\(L_{\omega_1,\omega}\) Theories

Incomplete Map of Non-elementary Classes

First Order Theories

\(L_{\omega_1,\omega}\) Theories

\(L_{\kappa^+,\omega}\) Theories

Incomplete Map of Non-elementary Classes

First Order Theories

\(L_{\omega_1,\omega}\) Theories

\(L_{\kappa^+,\omega}\) Theories

Abstract Elementary Classes

Incomplete Map of Non-elementary Classes

First Order Theories

\(L_{\omega_1,\omega}\) Theories

\(L_{\kappa^+,\omega}\) Theories

Abstract Elementary Classes

Tame

Incomplete Map of Non-elementary Classes

First Order Theories

\(L_{\omega_1,\omega}\) Theories

\(L_{\kappa^+,\omega}\) Theories

Abstract Elementary Classes

Universal, Homogeneous, Finitary

Non-elementary Classes

Tame

Abstract Elementary Classes

Classification Theory for

Tame

Abstract Elementary Classes

Classification Theory for

Tame

Categorical in high enough cardinality

Abstract Elementary Classes

Classification Theory for

Tame

Categorical in high enough cardinality

Stable

Abstract Elementary Classes

Classification Theory for

Tame

Categorical in high enough cardinality

Stable

Superstable

Test Question for
Classification Theory of AECs

First Order Morley's Theorem: Suppose \(T\) is a countable complete first order theory. If there exists an uncountable \(\lambda\) such that \(T\) is \(\lambda\) categorical, then \(T\) is \(\mu\)-categorical in all uncountable \(\mu\).

Test Question for
Classification Theory of AECs

Infinitary Logic Conjecture (Shelah, 1976): Suppose \(\psi\in L_{\omega_1,\omega}\) in a countable language. If there exists a \(\lambda\geq\beth_{\omega_1}\) such that \(\psi\) is \(\lambda\) categorical, then \(\psi\) is \(\mu\)-categorical in all \(\mu\geq \beth_{\omega_1}\).

Test Question for
Classification Theory of AECs

AEC Conjecture: Suppose \(\mathcal{K}\) is an AEC in a language of cardinality \(LS(\mathcal{K})\). If there exists a \(\lambda \geq \beth_{(2^{LS(\mathcal{K})})^+}\)\(^*\) such that \(\mathcal{K}\) is \(\lambda\)-categorical, then \(\mathcal{K}\) is \(\mu\)-categorical in all \(\mu\geq \beth_{(2^{LS(\mathcal{K})})^+}\).

\(^*\) This bound is smaller when \(LS(\mathcal{K})=\aleph_0\).

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a ordering, \(\prec\), satisfying:

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

If \(M\in\mathcal{K}\) and \(M\cong N\), then \(N\in\mathcal{K}\).

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

If \(M,N\in\mathcal{K}\) and \(M\prec N\),

then \(M\) is a submodel of \(N\).

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

If \(M\prec M^\), \(N\prec M^\), and \(M\subseteq N\),

then \(M\prec N\)

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

If \(M\prec M^\), \(N\prec M^\), and \(M\subseteq N\),

then \(M\prec N\)

\(M\)

\(M^*\)

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

If \(M\prec M^\), \(N\prec M^\), and \(M\subseteq N\),

then \(M\prec N\)

\(N\)

\(M^*\)

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

If \(M\prec M^\), \(N\prec M^\), and \(M\subseteq N\),

then \(M\prec N\)

\(M\)

\(N\)

\(M^*\)

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

If \(M\prec M^\), \(N\prec M^\), and \(M\subseteq N\),

then \(M\preceq N\).

\(M\)

\(N\)

\(M^*\)

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

\(A\)

\(M\)

A4. Löwenheim Skolem

There exists \(\lambda = LS(\mathcal{K})\) such that for every \(M\in\mathcal{K}\) and for every \(A\subseteq M\), there exists \(N\in\mathcal{K}\) so that \(N\prec M\) and \(\|N\|\leq |A|+\lambda\).

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

\(A\)

\(N\)

\(M\)

A4. Löwenheim Skolem

There exists \(\lambda=LS(\mathcal{K})\) such that for every \(M\in\mathcal{K}\) and for every \(A\subseteq M\), there exists \(N\in\mathcal{K}\) so that \(N\preceq M\) and \(\|N\|\leq |A|+\lambda\).

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5a. Union Conditions

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

\(M_0\)

\(M_i\)

\(M_{i+1}\)

A4. Löwenheim Skolem

If \(\langle M_i\in\mathcal{K}\mid i<\alpha\rangle\) is an \(\prec\)-increasing sequence, then \(\bigcup_{i<\alpha}M_i\in\mathcal{K}\) and for each \(i<\alpha\), \(M_i\prec \bigcup_{i<\alpha}M_i\).

A5a. Union Conditions

. . .

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

\(M_0\)

\(M_i\)

\(M_{i+1}\)

A4. Löwenheim Skolem

If \(\langle M_i\in\mathcal{K}\mid i<\alpha\rangle\) is an \(\prec\)-increasing sequence, then \(\bigcup_{i<\alpha}M_i\in\mathcal{K}\) and for each \(i<\alpha\), \(M_i\prec \bigcup_{i<\alpha}M_i\).

A5a. Union Conditions

. . .

\(\bigcup_{i<\alpha}M_{i}\)

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

\(M_i\)

A4. Löwenheim Skolem

If \(\langle M_i\in\mathcal{K}\mid i<\alpha\rangle\) is an \(\prec\)-increasing sequence, then \(\bigcup_{i<\alpha}M_i\in\mathcal{K}\) and for each \(i<\alpha\), \(M_i\prec \bigcup_{i<\alpha}M_i\).

A5a. Union Conditions

\(\bigcup_{i<\alpha}M_{i}\)

\(M_0\)

\(M_{i+1}\)

. . .

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5a.

A5b. Union Conditions

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

\(M_0\)

\(M_i\)

\(M_{i+1}\)

A4. Löwenheim Skolem

If \(\langle M_i\in\mathcal{K}\mid i<\alpha\rangle\) is an \(\prec\)-increasing sequence and \(N\) is such that for each \(i<\alpha\), \(M_i\prec N\), then \(\bigcup_{i<\alpha}M_i\prec N\).

. . .

A5a.

A5b. Union Conditions

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

\(M_0\)

\(M_i\)

\(M_{i+1}\)

A4. Löwenheim Skolem

If \(\langle M_i\in\mathcal{K}\mid i<\alpha\rangle\) is an \(\prec\)-increasing sequence and \(N\) is such that for each \(i<\alpha\), \(M_i\prec N\), then \(\bigcup_{i<\alpha}M_i\prec N\).

A5b. Union Conditions

. . .

\(N\)

A5a.

AEC Definition

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

\(M_0\)

\(M_i\)

\(M_{i+1}\)

A4. Löwenheim Skolem

If \(\langle M_i\in\mathcal{K}\mid i<\alpha\rangle\) is an \(\prec\)-increasing sequence and \(N\) is such that for each \(i<\alpha\), \(M_i\prec N\), then \(\bigcup_{i<\alpha}M_i\preceq N\).

A5b. Union Conditions

. . .

\(\bigcup_{i<\alpha}M_{i}\)

\(N\)

A5a.

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

Amalgamation Property

A5. Union Conditions

\(N\)

\(M_2\)

\(M_1\)

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

Amalgamation Property

A5. Union Conditions

\(N\)

\(M_2\)

\(M'\)

\(M_1\)

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

Amalgamation Property

A5. Union Conditions

\(N\)

\(M_2\)

\(M'\)

\(f\)

\(M_1\)

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

Amalgamation Property

A5. Union Conditions

\(N\)

\(M_2\)

\(M'\)

\(f\)

\(f(M_1)\)

\(M_1\)

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

Amalgamation Property

A5. Union Conditions

\(N\)

\(M_2\)

\(M'\)

\(f\)

\(f(M_1)\)

\(M_1\)

\(f\restriction N=id_N\)

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5. Union Conditions

Joint Embedding Property

\(M_1\)

\(M_2\)

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5. Union Conditions

Joint Embedding Property

\(M_1\)

\(M_2\)

\(M'\)

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5. Union Conditions

Joint Embedding Property

\(f_1(M_1)\)

\(f_2(M_2)\)

\(M'\)

\(f_1\)

\(f_2\)

\(M_1\)

\(M_2\)

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5. Union Conditions

\(M\)

No Maximal Models

JEP

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5. Union Conditions

\(M\)

No Maximal Models

\(M'\)

JEP

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5. Union Conditions

JEP

No Maximal Models

This implies the existence of a monster model.

\(M\)

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5. Union Conditions

JEP

No Maximal Models

This implies the existence of a monster model, \(\frak{C}\).

\(\frak{C}\)

\(M\)

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5. Union Conditions

JEP

No Maximal Models

\(M'\)

This implies the existence of a monster model, \(\frak{C}\).

\(\frak{C}\)

\(f\)

\(M\)

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5. Union Conditions

JEP

\(M\)

No Maximal Models

\(M'\)

This implies the existence of a monster model, \(\frak{C}\).

\(\frak{C}\)

\(f\)

\(f(M')\)

\(f\restriction M=id_M\)

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5. Union Conditions

JEP

\(A\)

No Maximal Models

This implies the existence of a monster model, \(\frak{C}\).

\(\frak{C}\)

Allowing us to define (Galois) types over models.

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5. Union Conditions

JEP

\(A\)

No Maximal Models

This implies the existence of a monster model, \(\frak{C}\).

\(\frak{C}\)

Allowing us to define (Galois) types.

\(tp(a/A)=tp(b/A)\)

\(a\)

\(b\)

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5. Union Conditions

JEP

\(A\)

No Maximal Models

This implies the existence of a monster model, \(\frak{C}\).

\(\frak{C}\)

Allowing us to define (Galois) types.

\(tp(a/A)=tp(b/A)\)

\(a\)

\(M_a\)

\(M_b\)

\(b\)

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5. Union Conditions

JEP

\(A\)

No Maximal Models

This implies the existence of a monster model, \(\frak{C}\).

\(\frak{C}\)

Allowing us to define (Galois) types.

\(tp(a/A)=tp(b/A)\) iff there exists \(f\) an automophism of \(\frak{C}\)

\(a\)

\(M_a\)

\(M_b\)

\(f\)

\(b\)

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5. Union Conditions

JEP

\(A\)

No Maximal Models

This implies the existence of a monster model, \(\frak{C}\).

\(\frak{C}\)

Allowing us to define (Galois) types.

\(tp(a/A)=tp(b/A)\) iff there exists \(f\) an automophism of \(\frak{C}\)
so that \(f\restriction A=id_A\) and

\(a\)

\(b\)

\(M_a\)

\(M_b\)

\(f\)

\(f\restriction A=id_A\)

Our Setting

\(\mathcal{K}\) is a class of models in a fixed language \(L(\mathcal{K})\) with a partial ordering, \(\prec\), satisfying:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

A5. Union Conditions

JEP

\(A\)

No Maximal Models

This implies the existence of a monster model, \(\frak{C}\).

\(\frak{C}\)

Allowing us to define (Galois) types.

\(tp(a/A)=tp(b/A)\) iff there exists \(f\) an automophism of \(\frak{C}\)
so that \(f\restriction A=id_A\) and \(f(a)=b\).

\(a\)

\(b=f(a)\)

\(M_a\)

\(M_b\)

\(f\)

\(f(M_a)\)

\(f\restriction A=id_A\)

Our Setting - Tame AECs

\(\mathcal{K}\) is an Abstract Elementary Class satisfying AP, JEP, and NMM.

A class is \(\chi\)-tame if for every \(N\in\mathcal{K}\) of cardinality \(\geq\chi\), and for every pair of types over \(N\), if \(q_a=tp(a/N)\neq tp(b/N)=q_b\) there exists \(A\subseteq N\) of cardinality \(<\chi\), so that \(q_a\restriction A\neq q_b\restriction A\).

\(\frak{C}\)

\(a\)

\(b\)

\(N\)

Our Setting - Tame AECs

\(\mathcal{K}\) is an Abstract Elementary Class satisfying AP, JEP, and NMM.

A class is \(\chi\)-tame if for every \(N\in\mathcal{K}\) of cardinality \(\geq\chi\), and for every pair of types over \(N\), if \(q_a=tp(a/N)\neq tp(b/N)=q_b\) there exists \(M\prec N\) of cardinality \(<\chi\), so that \(q_a\restriction M\neq q_b\restriction M\).

\(\frak{C}\)

\(a\)

\(N\)

In other words, if \(tp(a/M)=tp(b/M)\) for every \(M\subseteq N\) of cardinality \(<\chi\), then \(tp (a/N)=tp(b/N)\).

\(M\)

\(b\)

Our Setting - Tame AECs

\(\mathcal{K}\) is an Abstract Elementary Class satisfying AP, JEP, and NMM.

A class is \(\chi\)-tame if for every \(N\in\mathcal{K}\) of cardinality \(\geq\chi\), and for every pair of types over \(N\), if \(q_a=tp(a/N)\neq tp(b/N)=q_b\) there exists \(M\prec N\) of cardinality \(<\chi\), so that \(q_a\restriction M\neq q_b\restriction M\).

\(\frak{C}\)

\(a\)

\(b=f_M(a)\)

\(f_M(N)\)

In other words, if \(tp(a/M)=tp(b/M)\) for every \(M\prec N\) of cardinality \(<\chi\), then \(tp (a/N)=tp(b/N)\).

\(M\)

\(f_M\)

\(f_M\restriction M=id_M\)

\(N\)

Our Setting - Tame AECs

\(\mathcal{K}\) is an Abstract Elementary Class satisfying AP, JEP, and NMM.

A class is \(\chi\)-tame if for every \(N\in\mathcal{K}\) of cardinality \(\geq\chi\), and for every pair of types over \(N\), if \(q_a=tp(a/N)\neq tp(b/N)=q_b\) there exists \(M\prec N\) of cardinality \(<\chi\), so that \(q_a\restriction M\neq q_b\restriction M\).

\(\frak{C}\)

\(a\)

\(N\)

In other words, if \(tp(a/M)=tp(b/M)\) for every \(M\prec N\) of cardinality \(<\chi\), then \(tp (a/N)=tp(b/N)\).

\(M'\)

\(b\)

\(N\)

Our Setting - Tame AECs

\(\mathcal{K}\) is an Abstract Elementary Class satisfying AP, JEP, and NMM.

A class is \(\chi\)-tame if for every \(N\in\mathcal{K}\) of cardinality \(\geq\chi\), and for every pair of types over \(N\), if \(q_a=tp(a/N)\neq tp(b/N)=q_b\) there exists \(M\prec N\) of cardinality \(<\chi\), so that \(q_a\restriction M\neq q_b\restriction M\).

\(\frak{C}\)

\(a\)

\(b=f_{M'}(a)\)

\(f_{M'}(N)\)

In other words, if \(tp(a/M)=tp(b/M)\) for every \(M\prec N\) of cardinality \(<\chi\), then \(tp (a/N)=tp(b/N)\).

\(M'\)

\(f_{M'}\)

\(f_{M'}\restriction M'=id_{M'}\)

\(N\)

Our Setting - Tame AECs

\(\mathcal{K}\) is an Abstract Elementary Class satisfying AP, JEP, and NMM.

A class is \(\chi\)-tame if for every \(N\in\mathcal{K}\) of cardinality \(\geq\chi\), and for every pair of types over \(N\), if \(q_a=tp(a/N)\neq tp(b/N)=q_b\) there exists \(M\prec N\) of cardinality \(<\chi\), so that \(q_a\restriction M\neq q_b\restriction M\).

\(\frak{C}\)

\(a\)

\(N\)

In other words, if \(tp(a/M)=tp(b/M)\) for every \(M\prec N\) of cardinality \(<\chi\), then \(tp (a/N)=tp(b/N)\).

\(M''\)

\(b\)

\(N\)

Our Setting - Tame AECs

\(\mathcal{K}\) is an Abstract Elementary Class satisfying AP, JEP, and NMM.

A class is \(\chi\)-tame if for every \(N\in\mathcal{K}\) of cardinality \(\geq\chi\), and for every pair of types over \(N\), if \(q_a=tp(a/N)\neq tp(b/N)=q_b\) there exists \(M\prec N\) of cardinality \(<\chi\), so that \(q_a\restriction M\neq q_b\restriction M\).

\(\frak{C}\)

\(a\)

\(f_{M''}(N)\)

In other words, if \(tp(a/M)=tp(b/M)\) for every \(M\prec N\) of cardinality \(<\chi\), then \(tp (a/N)=tp(b/N)\).

\(M''\)

\(f_{M''}\)

\(b=f_{M''}(a)\)

\(f_{M''}\restriction M''=id_{M''}\)

\(N\)

Our Setting - Tame AECs

\(\mathcal{K}\) is an Abstract Elementary Class satisfying AP, JEP, and NMM.

A class is \(\chi\)-tame if for every \(N\in\mathcal{K}\) of cardinality \(\geq\chi\), and for every pair of types over \(N\), if \(q_a=tp(a/N)\neq tp(b/N)=q_b\) there exists \(M\prec N\) of cardinality \(<\chi\), so that \(q_a\restriction M\neq q_b\restriction M\).

\(\frak{C}\)

\(a\)

\(N\)

In other words, if \(tp(a/M)=tp(b/M)\) for every \(M\prec N\) of cardinality \(<\chi\), then \(tp (a/N)=tp(b/N)\).

\(b\)

Our Setting - Tame AECs

\(\mathcal{K}\) is an Abstract Elementary Class satisfying AP, JEP, and NMM.

Definition (Grossberg-V) A class is \(\chi\)-tame if for every \(N\in\mathcal{K}\) of cardinality \(\geq\chi\), and for every pair of types over \(N\), if \(q_a=tp(a/N)\neq tp(b/N)=q_b\) there exists \(M\prec N\) of cardinality \(<\chi\), so that \(q_a\restriction M\neq q_b\restriction M\).

\(\frak{C}\)

\(a\)

\(N\)

In other words, if \(tp(a/M)=tp(b/M)\) for every \(M\prec N\) of cardinality \(<\chi\), then \(tp (a/N)=tp(b/N)\).

\(b=f_N(a)\)

\(f_N\)

\(f_N\restriction N=id_N\)

Why study tameness?

Tame AECs are "Everywhere"

\(Mod(\psi)\) where \(\psi\in L_{\kappa,\omega}\) with \(\kappa\) strongly compact (Makkai-Shelah)
Homogeneous Classes
Finitary Classes
Quasi-minimal class axiomatizing Schanuel's Conjecture (Zilber)
Excellent classes (Kolesnikov-Grossberg)
Universal classes (Boney)
All AECs are tame iff there is class many almost strongly compact cardinals (Boney, Boney-Unger)

Tameness Informs Work on Shelah's Categoricity Conj.

\(^*\) This bound is smaller when \(LS(\mathcal{K})=\aleph_0\).

Until 2006 most related results

involved set theoretic assumptions.

Shelah's Categoricity Conjecture (1970s): Suppose \(\mathcal{K}\) is an AEC in a language of cardinality \(LS(\mathcal{K})\). If there exists a \(\lambda \geq \beth_{(2^{LS(\mathcal{K})})^+}\)\(^*\) such that \(\mathcal{K}\) is \(\lambda\)-categorical, then \(\mathcal{K}\) is \(\mu\)-categorical in all \(\mu\geq \beth_{(2^{LS(\mathcal{K})})^+}\).

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{\omega_1}<\mu<\lambda^+\).

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

\(\beth_{\omega_1}\)

categoricity in \(\lambda^+\)

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{\omega_1}<\mu<\lambda^+\).

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

\(\beth_{\omega_1}\)

categoricity in \(\mu\)

categoricity in \(\lambda^+\)

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{\omega_1}<\mu<\lambda^+\).

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

\(\beth_{\omega_1}\)

Upward Categoricity Transfer (Grossberg-V, 2006)

If \(\mathcal{K}\) is \(\chi\)-tame for some \(\chi\geq LS(\mathcal{K})\) and is categorical in some \(\lambda^+> LS(\mathcal{K})+\chi^+\), then it is categorical in all \(\mu\) satisfying \(\lambda^+\leq\mu\).

\(LS(\mathcal{K})\)

categoricity in \(\lambda^+\)

\(LS(\mathcal{K})+\chi^+\)

categoricity in \(\lambda^+\)

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{\omega_1}<\mu<\lambda^+\).

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

\(\beth_{\omega_1}\)

Upward Categoricity Transfer (Grossberg-V, 2006)

\(LS(\mathcal{K})\)

categoricity in \(\lambda^+\)

\(LS(\mathcal{K})+\chi^+\)

categoricity in \(\mu\)

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{\omega_1}<\mu<\lambda^+\).

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

\(\beth_{\omega_1}\)

Upward Categoricity Transfer (Grossberg-V, 2006)

\(LS(\mathcal{K})\)

categoricity in \(\lambda^+\)

Upward and Downward Bounds Improved (Vasey, 2017)

If \(\mathcal{K}\) is \(LS(\mathcal{K})\)-tame and is categorical in some \(\lambda^+>LS(\mathcal{K})\), then it is categorical in all \(\mu>min(\lambda,\beth_{(2^{LS(\mathcal{K})})^+})\).

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

\(LS(\mathcal{K})+\chi^+\)

categoricity in \(\lambda^+\)

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{\omega_1}<\mu<\lambda^+\).

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

\(\beth_{\omega_1}\)

Upward Categoricity Transfer (Grossberg-V, 2006)

\(LS(\mathcal{K})\)

categoricity in \(\lambda^+\)

Upward and Downward Bounds Improved (Vasey, 2017)

If \(\mathcal{K}\) is \(LS(\mathcal{K})\)-tame and is categorical in some \(\lambda^+>LS(\mathcal{K})\), then it is categorical in all \(\mu>min(\lambda,\beth_{(2^{LS(\mathcal{K})})^+})\).

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

\(LS(\mathcal{K})+\chi^+\)

categoricity in \(\mu\)

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{\omega_1}<\mu<\lambda^+\).

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

\(\beth_{\omega_1}\)

Upward Categoricity Transfer (Grossberg-V, 2006)

\(LS(\mathcal{K})\)

categoricity in \(\lambda^+\)

Upward and Downward Bounds Improved (Vasey, 2017)

If \(\mathcal{K}\) is \(LS(\mathcal{K})\)-tame and is categorical in some \(\lambda^+>LS(\mathcal{K})\), then it is categorical in all \(\mu>min(\lambda,\beth_{(2^{LS(\mathcal{K})})^+})\).

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

\(LS(\mathcal{K})+\chi^+\)

categoricity in \(\mu\)

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{\omega_1}<\mu<\lambda^+\).

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

\(\beth_{\omega_1}\)

Upward Categoricity Transfer (Grossberg-V, 2006)

\(LS(\mathcal{K})\)

categoricity in \(\lambda^+\)

Upward and Downward Bounds Improved (Vasey, 2017)

If \(\mathcal{K}\) is \(LS(\mathcal{K})\)-tame and is categorical in some \(\lambda^+>LS(\mathcal{K})\), then it is categorical in all \(\mu>min(\lambda,\beth_{(2^{LS(\mathcal{K})})^+})\).

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

\(LS(\mathcal{K})+\chi^+\)

categoricity in \(\lambda^+\)

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{\omega_1}<\mu<\lambda^+\).

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

\(\beth_{\omega_1}\)

Upward Categoricity Transfer (Grossberg-V, 2006)

\(LS(\mathcal{K})\)

categoricity in \(\mu\)

Upward and Downward Bounds Improved (Vasey, 2017)

If \(\mathcal{K}\) is \(LS(\mathcal{K})\)-tame and is categorical in some \(\lambda^+>LS(\mathcal{K})\), then it is categorical in all \(\mu>min(\lambda,\beth_{(2^{LS(\mathcal{K})})^+})\).

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

\(LS(\mathcal{K})+\chi^+\)

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{\omega_1}<\mu<\lambda^+\).

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

\(\beth_{\omega_1}\)

Upward Categoricity Transfer (Grossberg-V, 2006)

\(LS(\mathcal{K})\)

categoricity in \(\mu\)

Upward and Downward Bounds Improved (Vasey, 2017)

If \(\mathcal{K}\) is \(LS(\mathcal{K})\)-tame and is categorical in some \(\lambda^+>LS(\mathcal{K})\), then it is categorical in all \(\mu>min(\lambda,\beth_{(2^{LS(\mathcal{K})})^+})\).

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

\(LS(\mathcal{K})+\chi^+\)

\(\lambda^+\)

\(\chi\)-tame

AP, JEP, and NMM

If \(\mathcal{K}\) is categorical in some \(\lambda^+>\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\), then it is categorical in all \(\mu\) satisfying \(\beth_{\omega_1}<\mu<\lambda^+\).

Downward Categoricity Transfer (Shelah, 1999)

Let \(\mathcal{K}\) be an AEC satisfying AP, JEP, and NMM.

Upward Categoricity Transfer (Grossberg-V, 2006)

Upward and Downward Bounds Improved (Vasey, 2017)

If \(\mathcal{K}\) is \(LS(\mathcal{K})\)-tame and is categorical in some \(\lambda^+>LS(\mathcal{K})\), then it is categorical in all \(\mu>min(\lambda,\beth_{(2^{LS(\mathcal{K})})^+})\).

\(\lambda^+\)

\(\chi\)-tame

AP, JEP, and NMM

\(\lambda^+\)

\(\chi\)-tame

AP, JEP, and NMM

follow from categoricity

successor assumption removed in categoricity cardinal

follows from categoricity

What the downward/upward results leave open...

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

\(\beth_{\omega_1}\)

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

Categoricity for Universal Classes (Vasey 2017)

Let \(\mathcal{K}\) be a universal class with \(LS(\mathcal{K})=\aleph_0\).

\(\lambda^+\)

\(\chi\)-tame

AP, JEP, and NMM

follow from categoricity

successor assumption removed in categoricity cardinal

follows from categoricity

What the downward/upward results leave open...

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

\(\beth_{\omega_1}\)

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

\(\lambda^+\)

\(\chi\)-tame

AP, JEP, and NMM

follow from categoricity

successor assumption removed in categoricity cardinal

follows from categoricity

What the downward/upward results leave open...

Categoricity for Universal Classes (Vasey 2017)

Let \(\mathcal{K}\) be a universal class with \(LS(\mathcal{K})=\aleph_0\).

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

\(\beth_{\omega_1}\)

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

What the downward/upward results leave open...

\(\lambda^+\)

\(\chi\)-tame

AP, JEP, and NMM

follow from categoricity

successor assumption removed in categoricity cardinal

follows from categoricity

What the downward/upward results leave open...

Categoricity for Universal Classes (Vasey 2017)

Let \(\mathcal{K}\) be a universal class with \(LS(\mathcal{K})=\aleph_0\).

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

\(\beth_{\omega_1}\)

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

\(\beth_{\omega_1}\)

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

What the downward/upward results leave open...

Let \(\mathcal{K}\) be a universal class with \(LS(\mathcal{K})=\aleph_0\). If \(\mathcal{K}\) is categorical in some \(\lambda>\beth_{\beth_{\omega_1}}\) then \(\mathcal{K}\) is categorical in all \(\lambda>\beth_{\beth_{\omega_1}}\).

\(\lambda^+\)

\(\chi\)-tame

AP, JEP, and NMM

follow from categoricity

successor assumption removed in categoricity cardinal

follows from categoricity

What the downward/upward results leave open...

Categoricity for Universal Classes (Vasey 2017)

\(\beth_{\beth_{(2^{LS(\mathcal{K})})^+}}\)

\(\beth_{\omega_1}\)

\(\beth_{(2^{LS(\mathcal{K})})^+}\)

Limit Models

\(M_0\)

\(M_i\)

\(M_{i+1}\)

. . .

Definition (Shelah). If there exists a \(\prec\)-increasing sequence \(\langle M_i\in\mathcal{K}_\mu\mid i<\theta\rangle\) so that for each \(i<\theta\), \(M_{i+1}\) is universal over \(M_i\), then \(\bigcup_{i<\theta}M_i\) is a \((\mu,\theta)\)-limit model.

\(\bigcup_{i<\theta}M_{i}\)

Limit Models

\(M_0\)

\(M_i\)

\(M_{i+1}\)

. . .

\(M'\)

\(\bigcup_{i<\theta}M_{i}\)

Limit Models

\(M_0\)

\(M_i\)

\(M_{i+1}\)

. . .

\(M'\)

\(f\)

\(f(M')\)

\(\bigcup_{i<\theta}M_{i}\)

Limit Models

\(M_0\)

\(M_i\)

\(M_{i+1}\)

. . .

\(\bigcup_{i<\theta}M_{i}\)

\(M'\)

\(f\)

\(f(M')\)

\(f\restriction M_i=id_{M_i}\)

Uniqueness of Limit Models

Suppose that \(\alpha\) and \(\theta\) are limit ordinals \(<\mu^+\) and \(M\in\mathcal{K}_\mu\).

Let \(\mathcal{K}\) be an AEC and fix \(\mu>LS(\mathcal{K})\).

Uniqueness question: Under what non-trivial conditions can we conclude that if \(M^\alpha\) is a \((\mu,\alpha)\)-limit model over \(M\) and \(M^\theta\) is a \((\mu,\theta)\)-limit model over \(M\), then there exists \(f:M^\alpha\cong M^\theta\) with \(f\restriction M=id_M\)?

\(M=M^\alpha_0\)

\(M^\alpha_{i+1}\)

\(M^\alpha_{i}\)

\(M^\alpha=\bigcup_{i<\alpha} M^\alpha_i\)

Uniqueness of Limit Models

Suppose that \(\alpha\) and \(\theta\) are limit ordinals \(<\mu^+\) and \(M\in\mathcal{K}_\mu\).

Let \(\mathcal{K}\) be an AEC and fix \(\mu>LS(\mathcal{K})\).

\(M=M^\theta_0=M^\alpha_0\)

\(M^\theta_i\)

\(M^\theta_{i+1}\)

. . .

\(\bigcup_{i<\theta}M^\theta_{i}\)

\(M^\alpha_{i+1}\)

\(M^\alpha_{i}\)

\(M^\alpha=\bigcup_{i<\alpha} M^\alpha_i\)

Uniqueness of Limit Models

Suppose that \(\alpha\) and \(\theta\) are limit ordinals \(<\mu^+\) and \(M\in\mathcal{K}_\mu\).

Let \(\mathcal{K}\) be an AEC and fix \(\mu>LS(\mathcal{K})\).

\(M=M^\theta_0=M^\alpha_0\)

\(M^\theta_i\)

\(M^\theta_{i+1}\)

. . .

\(\bigcup_{i<\theta}M^\theta_{i}\)

\(M^\alpha_{i+1}\)

\(M^\alpha_{i}\)

\(M^\alpha=\bigcup_{i<\alpha} M^\alpha_i\)

\(f\)

Uniqueness of Limit Models

Suppose that \(\alpha\) and \(\theta\) are limit ordinals \(<\mu^+\) and \(M\in\mathcal{K}_\mu\).

Let \(\mathcal{K}\) be an AEC with AP, JEP, and NMM, and fix \(\mu>LS(\mathcal{K})\).

Case \(cf(\alpha)=cf(\theta)\): Back and forth construction produces \(f\).

Case \(cf(\alpha)\neq cf(\theta)\): Answer seems related to "superstability".

Some Answers in Categorical Settings

Answer #1: Infinitary Logic

If \(\mathcal{K}\) is categorical in \(\lambda>LS(\mathcal{K})\) and \(\mathcal{K}=Mod(\psi)\) where \(\psi\in L_{\kappa,\omega}\)

for \(\kappa\) is strongly compact (Makkai-Shelah, 1990)
for \(\kappa\) is measurable and \(\mu<\lambda\) (Kolman-Shelah, 1996)

Some Answers in Categorical Settings

Answer # 2: AECs with set theoretic assumptions

If \(\mathcal{K}\) is categorical in \(\lambda>LS(\mathcal{K})\) and has NMM.

(Shelah-Villaveces, 1999) attempt uses GCH and diamond.
- GCH implies density of amalgamation bases
- Limit models are amalgamation bases and dense
- Categoricity implies no long splitting chains (Boney-Grossberg-V-Vasey, 2017).
- No long splitting chains implies uniqueness of limit models.

Some Answers in Categorical Settings

Answer # 2: AECs with set theoretic assumptions

If \(\mathcal{K}\) is categorical in \(\lambda>LS(\mathcal{K})\) and has NMM.

(Shelah-Villaveces, 1999) attempt uses GCH and diamond.
- GCH implies density of amalgamation bases
- Limit models are amalgamation bases and dense
- Categoricity implies no long splitting chains (Boney-Grossberg-V-Vasey, 2017).
- No long splitting chains implies uniqueness of limit models.

Some Answers in Categorical Settings

Answer # 2: AECs with set theoretic assumptions

If \(\mathcal{K}\) is categorical in \(\lambda>LS(\mathcal{K})\) and has NMM.

(Shelah-Villaveces, 1999) attempt uses GCH and diamond.
- GCH implies density of amalgamation bases
- Limit models are amalgamation bases and dense (V 2006)
- Categoricity implies no long splitting chains (Boney-Grossberg-V-Vasey, 2017).
- No long splitting chains implies uniqueness of limit models.

Some Answers in Categorical Settings

Answer # 2: AECs with set theoretic assumptions

If \(\mathcal{K}\) is categorical in \(\lambda>LS(\mathcal{K})\) and has NMM.

(Shelah-Villaveces, 1999) attempt uses GCH and diamond.
- GCH implies density of amalgamation bases
- Limit models are amalgamation bases and dense (V 2006)
- Categoricity implies no long splitting chains (Boney-Grossberg-V-Vasey, 2017).
- No long splitting chains implies uniqueness of limit models.

Some Answers in Categorical Settings

Answer # 2: AECs with set theoretic assumptions

If \(\mathcal{K}\) is categorical in \(\lambda>LS(\mathcal{K})\) and has NMM.

(Shelah-Villaveces, 1999) attempt uses GCH and diamond.
- GCH implies density of amalgamation bases
- Limit models are amalgamation bases and dense (V 2006)
- Categoricity implies no long splitting chains (Boney-Grossberg-V-Vasey, 2017).
- No long splitting chains implies uniqueness of limit models (and \(\mu\)-symmetry.) (V, 2006, 2013, 2016)

Some Answers in Categorical Settings

Answer #3: AECs with no set theoretic assumptions

If \(\mathcal{K}\) is categorical in \(\lambda\) sufficiently large and \(\mathcal{K}\) satisfies AP and JEP, then \(\mathcal{K}\) has uniqueness of limit models of cardinality \(\mu\) for \(\mu\geq LS(\mathcal{K})\). (Vasey-V, 2017)

Some Answers in Superstable Settings

If \(\mathcal{K}\) is \(\mu\)-superstable and satisfies AP, JEP, NMM and is tame and stable in a proper class of cardinals (Grossberg-Vasey, 2017)

If \(\mathcal{K}\) is \(\mu\)-superstable and satisfies AP, JEP, NMM and satisfies \(\mu\)-symmetry (V., 2016).

Answer # 4: in Superstable Settings:

Focusing on Superstability

If \(\mathcal{K}\) is \(\mu\)- and \(\mu^+\)-superstable and satisfies AP, JEP, NMM

\(\mu^+\)-symmetry

(Vasey-V., 2017)

\(\mu\)-symmetry

Uniqueness of limit models of cardinality \(\mu^+\)

Union of increasing chain of \(\mu^+\)-saturated models is saturated

Uniqueness of limit models of cardinality \(\mu\)

(V., 2016b)

(V., 2016a)

circa 2007

Abstract Elementary Classes

Classification Theory for

Tame

Categorical in high enough cardinality

Stable

Superstable

Tame Abstract Elementary Classes

Classification Theory for

Tame

Categorical in high enough cardinality

Stable

Superstable

circa 2007

Superstability in Tame AECs

Suppose \(\mathcal{K}\) satisfies the AP, JEP, and NMM and is tame. If \(\mathcal{K}\) is stable in unboundedly many cardinals, then TFAE:

For \(\lambda\) high enough, \(\mathcal{K}\) has uniqueness of limit models of cardinality \(\lambda\).
For \(\lambda\) high enough, \(\mathcal{K}\) is \(\lambda\)-superstable.
For \(\lambda\) high enough, the union of a chain of \(\lambda\)-saturated models in \(\lambda\) saturated.
There is a \(\theta\) so that for \(\lambda\) high enough, \(\mathcal{K}\) is \((\lambda,\theta)\)-solvable.
For \(\lambda\) high enough, there is \(\kappa=\kappa_\lambda\leq\lambda\) so that there is a good \(\lambda\)-frame on \(\mathcal{K}^{\kappa-sat}_\lambda\).
For \(\lambda\) high enough, \(\mathcal{K}\) has a superlimit model of cardinality \(\lambda\).

See (Boney-Vasey, 2017) for an exposition.

Limit Models without Superstabilty

Let \(\mathcal{K}\) be an AEC with AP, JEP, and NMM. \(M\in\mathcal{K_\lambda}\) is \(\lambda\)-saturated iff \(M\) is a \((\lambda,\lambda)\)-limit model (Grossberg-Vasey, 2017).

For \(\mathcal{K}\) an AEC with AP, JEP, and NMM. If \(\mathcal{K}\) is \(\mu\)-stable, has weak continuity of \(\mu\)-splitting, and satisfies \(\mu\)-symmetry, then there is a \(\kappa\) which for \(\alpha_l\) \((l=1,2)\) of cofinality \(>\kappa\), then \((\mu,\alpha_1)\) limit models are isomorphic to \((\mu,\alpha_2)\)-limit models. (Boney-V., nd)

Let \(\mathcal{K}\) be the class of torsion free Abelian groups with the pure subgroup relation. If \(G\in\mathcal{K}\) is a \((\lambda,\alpha)\)-limit model, then

for \(cf(\alpha)\) uncountable, \(G\cong (\bigoplus_\lambda\mathbb{Q})\bigoplus\Pi_{p\; prime}\overline{(\bigoplus_\lambda\mathbb{Z}_{(p)})}\)
for \(cf(\alpha)\) countable, \(G\) is not algebraically compact
(Mazari-Armida, 2018).

Uniqueness of Limit Models in Algebraic Settings

Given a ring \(R\), let \(\mathcal{K}\) be the set of left \(R\)-modules.

\(R\) is left Noetherian \(\Leftrightarrow\) \(\mathcal{K}\) is superstable \(\Leftrightarrow\)

Given a ring \(R\), let \(\mathcal{K}\) be the set of flat left \(R\)-modules with pure embeddings.

\(R\) is left perfect \(\Leftrightarrow\) \(\mathcal{K}\) is superstable \(\Leftrightarrow\)

Let \(\mathcal{K}\) be the class of Abelian groups with the subgroup relation. Then \(\mathcal{K}\) has uniqueness of limit models in every infinite cardinality (Mazari-Armida, 2018).

Limit models are unique in some cardinality \(\lambda>(|R|+\aleph_0)^+\) (Mazari-Armida, 2021).

Limit models are unique in all cardinalities \(\lambda\geq(|R|+\aleph_0)^+\) (Mazari-Armida, 2020)