Abstract independence relations in neostability theory
Alberto Miguel-Gómez
TL;DR
The paper develops a unifying framework that treats witnessing (Kim's Lemma) as a relation between abstract independence relations, extending relative Kim-independence to arbitrary independence notions. It introduces GUWP (and related variants) to capture witnessing across theories such as simplicity, $NTP_2$, and $NSOP_1$, and proves core connections among witnessing, symmetry, chain local character, and independence theorems. By deriving NSOP$_1$–BTP dichotomies and showing that many NSOP$_4$ examples fall into the BTP class, the results provide semantic unification and broad applicability across neostability theories. The work clarifies when witnessing implies classical independence properties, and offers a cohesive toolkit to study Kim-type independence in diverse settings with full existence and monotonicity assumptions.
Abstract
We develop a framework, in the style of Adler, for interpreting the notion of "witnessing" that has appeared (usually as a variant of Kim's Lemma) in different areas of neostability theory as a binary relation between abstract independence relations. This involves extending the relativisations of Kim-independence and Conant-independence due to Mutchnik to arbitrary independence relations. After developing this framework, we show that several results from simplicity, $\text{NTP}_2$, $\text{NSOP}_1$, and beyond follow as instances of general theorems for abstract independence relations. In particular, we prove the equivalence between witnessing and symmetry and the implications from this notion to chain local character and the weak independence theorem, and recover some partial converses. Finally, we use this framework to prove a dichotomy between $\text{NSOP}_1$ and Kruckman and Ramsey's $\text{BTP}$ that applies to most known $\text{NSOP}_4$ examples in the literature.