Tame abstract elementary class

In model theory, a discipline within the field of mathematical logic, a tame abstract elementary class is an abstract elementary class (AEC) which satisfies a locality property for types called tameness. Even though it appears implicitly in earlier work of Shelah, tameness as a property of AEC was first isolated by Grossberg and VanDieren,^[1] who observed that tame AECs were much easier to handle than general AECs.

Definition

Let K be an AEC with joint embedding, amalgamation, and no maximal models. Just like in first-order model theory, this implies K has a universal model-homogeneous monster model $\mathfrak{C}$. Working inside $\mathfrak{C}$, we can define a semantic notion of types by specifying that two elements a and b have the same type over some base model $M$ if there is an automorphism of the monster model sending a to b fixing $M$ pointwise (note that types can be defined in a similar manner without using a monster model^[2]). Such types are called Galois types.

One can ask for such types to be determined by their restriction on a small domain. This gives rise to the notion of tameness:

• An AEC $K$ is tame if there exists a cardinal $\kappa$ such that any two distinct Galois types are already distinct on a submodel of their domain of size $\le \kappa$. When we want to emphasize $\kappa$, we say $K$ is $\kappa$-tame.

Tame AECs are usually also assumed to satisfy amalgamation.

Discussion and motivation

While (without the existence of large cardinals) there are examples of non-tame AECs,^[3] most of the known natural examples are tame.^[4] In addition, the following sufficient conditions for a class to be tame are known:

• Tameness is a large cardinal axiom:^[5] There are class-many weakly strongly compact cardinals iff any abstract elementary class is tame.
• Some tameness follows from categoricity:^[6] If an AEC with amalgamation is categorical in a cardinal $\lambda$ of high-enough cofinality, then tameness holds for types over saturated models of size less than $\lambda$.

Many results in the model theory of (general) AECs assume weak forms of the Generalized continuum hypothesis and rely on sophisticated combinatorial set-theoretic arguments.^[7] On the other hand, the model theory of tame AECs is much easier to develop, as evidenced by the results presented below.

Results

The following are some important results about tame AECs.

• Upward categoricity transfer:^[8] A $\kappa$-tame AEC with amalgamation that is categorical in some successor $\lambda \ge \operatorname{LS}(K)^{++} + \kappa^+$ (i.e. has exactly one model of size $\lambda$ up to isomorphism) is categorical in all $\mu \ge \lambda$.
• Upward stability transfer:^[9] A $\kappa$-tame AEC with amalgamation that is stable in a cardinal $\lambda \ge \kappa$ is stable in $\lambda^+$ and in every infinite $\mu$ such that $\mu^\lambda = \mu$.
• Tameness can be seen as a topological separation principle: ^[10] An AEC with amalgamation is tame if and only if an appropriate topology on the set of Galois types is Hausdorff.
• Tameness and categoricity imply there is a forking notion: ^[11] A $\kappa$-tame AEC with amalgamation that is categorical in a cardinal $\lambda$ of cofinality greater than or equal to $\kappa$ has a good frame: a forking-like notion for types of singletons (in particular, it is stable in all cardinals). This gives rise to a well-behaved notion of dimension.

Notes

1. Grossberg & VanDieren 2006a.
2. Shelah 2009, Definition II.1.9.
3. Baldwin & Shelah 2008.
4. See the discussion in the introduction of Grossberg & VanDieren 2006a.
5. Boney 2014, Theorem 1.3.
6. Shelah 1999, Main claim 2.3 (9.2 in the online version).
7. See for example many of the hard theorems of Shelah's book (Shelah 2009).
8. Grossberg & VanDieren 2006b.
9. See Baldwin, Kueker & VanDieren 2006, Theorem 4.5 for the first result and Grossberg & VanDieren 2006a for the second.
10. Lieberman 2011, Proposition 4.1.
11. See Vasey 2014 for the first result, and Boney & Vasey 2014, Corollary 6.10.5 for the result on dimension.

References

• Shelah, Saharon (1999), "Categoricity for abstract classes with amalgamation" (PDF), Annals of Pure and Applied Logic 98 (1): 261–294, doi:10.1016/s0168-0072(98)00016-5
• Grossberg, Rami (2002), "Classification theory for abstract elementary classes" (PDF), Contemporary mathematics 302: 165–204, doi:10.1090/conm/302/05080
• Grossberg, Rami; VanDieren, Monica (2006a), "Galois-stability for tame abstract elementary classes" (PDF), Journal of Mathematical Logic 6 (1): 25–49, doi:10.1142/s0219061306000487
• Grossberg, Rami; VanDieren, Monica (2006b), "Categoricity from one successor cardinal in tame abstract elementary classes" (PDF), Journal of Mathematical Logic 6: 181–201, doi:10.1142/s0219061306000554
• Baldwin, John T.; Kueker, David; VanDieren, Monica (2006), "Upward stability transfer for tame abstract elementary classes" (PDF), Notre Dame Journal of Formal Logic 47 (2): 291–298, doi:10.1305/ndjfl/1153858652
• Baldwin, John T.; Shelah, Saharon (2008), "Examples of non-locality" (PDF), The Journal of Symbolic Logic 73: 765–782, doi:10.2178/jsl/1230396746
• Shelah, Saharon (2009), Classification theory for elementary abstract classes, Studies in Logic (London) 18, College Publications, London, ISBN 978-1-904987-71-0
• Baldwin, John T. (2009), Categoricity, University Lecture Series 50, American Mathematical Society, ISBN 978-0821848937
• Lieberman, Michael J. (2011), "A topology for Galois types in abstract elementary classes", Mathematical Logic Quarterly 57 (2): 204–216, doi:10.1002/malq.200910132
• Boney, Will (2014). "Tameness from large cardinal axioms". arXiv:1303.0550v4.
• Boney, Willl; Unger Spencer (2015), "Large Cardinal Axioms from Tameness in AECs" arXiv:1509.01191v2.
• Vasey, Sebastien (2014). "Forking and superstability in tame AECs". arXiv:1405.7443v2.
• Boney, Will; Vasey, Sebastien (2014). "Tameness and frames revisited". arXiv:1406.5980v4.