Local character of Kim-independence

TL;DR

This paper characterizes NSOP$_{1}$ theories through a local character property of Kim-independence, introducing a new dual local-character phenomenon specific to these theories.

Contribution

It establishes a precise equivalence between NSOP$_{1}$ and a local character property of Kim-independence, and introduces the concept of dual local-character.

Findings

Kim-independence satisfies local character in NSOP$_{1}$ theories

The collection of substructures where a type does not Kim-fork forms a club

Introduction of dual local-character phenomenon in NSOP$_{1}$ theories

Abstract

We show that NSOP$_{1}$ theories are exactly the theories in which Kim-independence satisfies a form of local character. In particular, we show that if $T$ is NSOP$_{1}$, $M\models T$, and $p$ is a type over $M$, then the collection of elementary substructures of size $\left|T\right|$ over which $p$ does not Kim-fork is a club of $\left[M\right]^{\left|T\right|}$ and that this characterizes NSOP$_{1}$. We also present a new phenomenon we call dual local-character for Kim-independence in NSOP$_{1}$-theories.