Abstract elementary class

In model theory, a discipline within mathematical logic, an abstract elementary class, or AEC for short, is a class of models with a partial order similar to the relation of an elementary substructure of an elementary class in first-order model theory. They were introduced by Saharon Shelah.
Definition.
formula_1, for formula_2 a class of structures in some language formula_3, is an AEC if it has the following properties:
Note that we usually do not care about the models of size less than the Löwenheim–Skolem number and often assume that there are none (we will adopt this convention in this article). This is justified since we can always remove all such models from an AEC without influencing its structure above the Löwenheim–Skolem number.
A formula_2-embedding is a map formula_39 for formula_40 such that formula_41 and formula_42 is an isomorphism from formula_7 onto formula_44. If formula_2 is clear from context, we omit it.
Examples.
The following are examples of abstract elementary classes:
Common assumptions.
AECs are very general objects and one usually make some of the assumptions below when studying them:
Note that in elementary classes, joint embedding holds whenever the theory is complete, while amalgamation and no maximal models are well-known consequences of the compactness theorem. These three assumptions allow us to build a universal model-homogeneous monster model formula_67, exactly as in the elementary case.
Another assumption that one can make is tameness.
Shelah's categoricity conjecture.
Shelah introduced AECs to provide a uniform framework in which to generalize first-order classification theory. Classification theory started with Morley's categoricity theorem, so it is natural to ask whether a similar result holds in AECs. This is Shelah's eventual categoricity conjecture. It states that there should be a Hanf number for categoricity:
For every AEC "K" there should be a cardinal formula_36 depending only on formula_35 such that if "K" is categorical in "some" formula_70 (i.e. "K" has exactly one (up to isomorphism) model of size formula_71), then "K" is categorical in formula_72 for "all" formula_73.
Shelah also has several stronger conjectures: The threshold cardinal for categoricity is the Hanf number of pseudoelementary classes in a language of cardinality LS(K). More specifically when the class is in a countable language and axiomaziable by an formula_74 sentence the threshold number for categoricity is formula_75. This conjecture dates back to 1976.
Several approximations have been published (see for example the results section below), assuming set-theoretic assumptions (such as the existence of large cardinals or variations of the generalized continuum hypothesis), or model-theoretic assumptions (such as amalgamation or tameness). As of 2014, the original conjecture remains open.
Results.
The following are some important results about AECs. Except for the last, all results are due to Shelah.