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.