Remarks on convergence of Morley sequences

TL;DR

This paper refines existing results on Morley sequences convergence, introduces the concept of eventual NIP, and provides new characterizations of generically stable types, advancing the understanding of model-theoretic stability and convergence properties.

Contribution

It improves the understanding of Morley sequence convergence, introduces eventual NIP, and offers new characterizations of generically stable types in countable theories.

Findings

Refined equivalences of convergent Morley sequences.

Introduced the notion of eventual NIP.

Provided new characterizations of generically stable types.

Abstract

We refine results of Gannon [G21, Theorem 4.7] and Simon [S15a, Lemma 2.8] on equivalences of convergent Morley sequences. We then introduce the notion of eventual $NIP$, as a property of a model, and give a variant of [KP18, Corollary 2.2]. Finally, we give new characterizations of generically stable types (for countable theories) and reinforce the main result of Pillay [P18] on the model-theoretic meaning of Grothendieck's double limit theorem.