Tame abstract elementary class

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 ${\displaystyle {\mathfrak {C}}}$. Working inside ${\displaystyle {\mathfrak {C}}}$, we can define a semantic notion of types by specifying that two elements a and b have the same type over some base model ${\displaystyle M}$ if there is an automorphism of the monster model sending a to b fixing ${\displaystyle M}$ 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 ${\displaystyle K}$ is tame if there exists a cardinal ${\displaystyle \kappa }$ such that any two distinct Galois types are already distinct on a submodel of their domain of size ${\displaystyle \leq \kappa }$. When we want to emphasize ${\displaystyle \kappa }$, we say ${\displaystyle K}$ is ${\displaystyle \kappa }$-tame.

Tame AECs are usually also assumed to satisfy amalgamation.

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 almost 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 ${\displaystyle \lambda }$ of high-enough cofinality, then tameness holds for types over saturated models of size less than ${\displaystyle \lambda }$.
• Conjecture 1.5 in [7]: If K is categorical in some λ ≥ Hanf(K) then there exists χ < Hanf(K) such that K is χ-tame.

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.[8] On the other hand, the model theory of tame AECs is much easier to develop, as evidenced by the results presented below.

The following are some important results about tame AECs.

• Upward categoricity transfer:[9] A ${\displaystyle \kappa }$-tame AEC with amalgamation that is categorical in some successor ${\displaystyle \lambda \geq \operatorname {LS} (K)^{++}+\kappa ^{+}}$ (i.e. has exactly one model of size ${\displaystyle \lambda }$ up to isomorphism) is categorical in all ${\displaystyle \mu \geq \lambda }$.
• Upward stability transfer:[10] A ${\displaystyle \kappa }$-tame AEC with amalgamation that is stable in a cardinal ${\displaystyle \lambda \geq \kappa }$ is stable in ${\displaystyle \lambda ^{+}}$ and in every infinite ${\displaystyle \mu }$ such that ${\displaystyle \mu ^{\lambda }=\mu }$.
• Tameness can be seen as a topological separation principle:[11] 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:[12] A ${\displaystyle \kappa }$-tame AEC with amalgamation that is categorical in a cardinal ${\displaystyle \lambda }$ of cofinality greater than or equal to ${\displaystyle \kappa }$ 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.

• Shelah, Saharon (1999), "Categoricity for abstract classes with amalgamation" (PDF), Annals of Pure and Applied Logic, 98 (1): 261–294, arXiv:math/9809197, doi:10.1016/s0168-0072(98)00016-5, S2CID 27872122
• Grossberg, Rami (2002), "Classification theory for abstract elementary classes" (PDF), Logic and algebra, Contemporary Mathematics, vol. 302, Providence, RI: American Mathematical Society, pp. 165–204, doi:10.1090/conm/302/05080, MR 1928390
• Grossberg, Rami; VanDieren, Monica (2006a), "Galois-stability for tame abstract elementary classes" (PDF), Journal of Mathematical Logic, 6 (1): 25–49, arXiv:math/0509535, doi:10.1142/s0219061306000487, S2CID 15621767
• Grossberg, Rami; VanDieren, Monica (2006b), "Categoricity from one successor cardinal in tame abstract elementary classes" (PDF), Journal of Mathematical Logic, 6 (2): 181–201, arXiv:math/0510004, doi:10.1142/s0219061306000554, S2CID 16930649
• 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, S2CID 5770095
• Baldwin, John T.; Shelah, Saharon (2008), "Examples of non-locality" (PDF), The Journal of Symbolic Logic, 73 (3): 765–782, doi:10.2178/jsl/1230396746, S2CID 7276664
• Shelah, Saharon (2009), Classification theory for elementary abstract classes, Studies in Logic (London), vol. 18, College Publications, London, ISBN 978-1-904987-71-0
• Baldwin, John T. (2009), Categoricity, University Lecture Series, vol. 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 [math.LO].
• Boney, Will; 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 [math.LO].
• Boney, Will; Vasey, Sebastien (2014). "Tameness and frames revisited". arXiv:1406.5980v4 [math.LO].