Abstract independence relations in neostability theory

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.