Let 𝒦 be the class of atomic models of a countable first order theory. 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 properties, 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 ω (and is not achieved). We illustrate the theory with the example of free groups, and Zilber’s pseudo analytic structures.
"Categoricity and U-rank in excellent classes." J. Symbolic Logic 68 (4) 1317 - 1336, December 2003. https://doi.org/10.2178/jsl/1067620189