Let K be the class of atomic models of a countable first order the- ory. We prove that if K is excellent and categorical in some uncountable cardinal, then each model is prime and minimal over the basis of a definable pregeometry given by a quasiminimal set. This implies that K is categorical in all uncountable cardinals. We also introduce a U-rank to measure the complexity of complete types over models. We prove that the U-rank has the usual additivity proper- ties, that quasiminimal types have U-rank 1, and that the U-rank of any type is finite in the uncountably categorical, excellent case. However, in contrast to the first order case, the supremum of the U-rank over all types may be ! (and is not achieved). We illustrate the theory with the example of free groups, and Zilber's pseudo analytic structures. A class of mathematical structures K is categorical in some cardinal ‚ if all the structures in K of size ‚ are isomorphic. The problem of categoricity can be roughly phrased as follows: Suppose K is categorical in some cardinal(s), is K also categorical in other (all) cardinals? The classical problem, when K is the class of models of a first order theory, has been a driving force in first order model theory, and it is difficult to overestimate the impact of Morley's theorem (Mo), Baldwin-Lachlan's Theorem (BaLa), and Shelah's generalisation to uncountable languages (Sh70) on its development. The present paper is concerned with the categoricity of classes of models which may not be axiomatisable in first order logic. There are several natural extensions of first order logic, many of which are equivalent for this problem. We will focus on classes of models of a first order theory T omitting a prescribed set of types i. There are two extreme cases: Wheni is empty; this is the first order case. When i is the set of nonisolated types; this is the atomic case, the class of models omitting all nonisolated types is the class of atomic models of T. For simplicity, and without real loss of generality, we consider the atomic case; in Remark 1.4 we explain how to develop excellence for some more general i. In the early 1970s, Keisler (Ke) and Shelah (Sh3) independently proved that if a class K of atomic models is categorical in some uncountable cardinal,
['Olivier Lessmann']
Categoricity and U-rank in excellent classes