Internality in Families

In geometric stability theory, internality plays a central role, as a tool to explore the fine structure of definable sets. Studying internal types, one often encounters uniformly defined families of internal types. This thesis is mainly concerned with the study of such families, both from an abstract model-theoretic perspective and in concrete examples.

In the first part, we generalize one of the fundamental tools of geometric stability theory, type-definable binding groups, to certain families of internal types, which we call relatively internal. We obtain, instead of a group, a type-definable groupoid, as well as simplicial data, encoding structural properties of the family.

In the second part, we introduce a new strengthening of internality, called uniform internality. We expose its connection with the previously constructed groupoids, and prove it is a strengthening of preserving internality, a notion previously introduced by Moosa. We then explore examples in differentially closed fields and compact complex manifolds.

In the last part, we study a structural feature of stable theories, called the canonical base property. We prove that Hrushovski, Palacín and Pillay's counterexample does not transfer to positive characteristic. Elaborating on the counterexample, we also provide an abstract configuration violating the canonical base property.