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. Working inside, we can define a semantic notion of types by specifying that two elements a and b have the same type over some base model if there is an automorphism of the monster model sending a to b fixing pointwise. 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 is tame if there exists a cardinal such that any two distinct Galois types are already distinct on a submodel of their domain of size. When we want to emphasize, we say is -tame.
Tame AECs are usually also assumed to satisfy amalgamation.
Discussion and motivation
While there are examples of non-tame AECs, most of the known natural examples are tame. In addition, the following sufficient conditions for a class to be tame are known:
Tameness is a large cardinal axiom: There are class-many almost strongly compact cardinals iff any abstract elementary class is tame.
Some tameness follows from categoricity: If an AEC with amalgamation is categorical in a cardinal of high-enough cofinality, then tameness holds for types over saturated models of size less than.
Conjecture 1.5 in : If K is categorical in some λ ≥ Hanf then there exists χ < Hanf such that K is χ-tame.
Many results in the model theory of AECs assume weak forms of the Generalized continuum hypothesis and rely on sophisticated combinatorial set-theoretic arguments. 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: A -tame AEC with amalgamation that is categorical in some successor is categorical in all.
Upward stability transfer: A -tame AEC with amalgamation that is stable in a cardinal is stable in and in every infinite such that.
Tameness and categoricity imply there is a forking notion: A -tame AEC with amalgamation that is categorical in a cardinal of cofinality greater than or equal to has a good frame: a forking-like notion for types of singletons. This gives rise to a well-behaved notion of dimension.
