# Transitivity of Kim-independence

## TL;DR

This paper establishes that Kim-independence satisfies transitivity in NSOP$_{1}$ theories and characterizes witnesses to Kim-dividing, providing new insights and resolving open questions in the model theory of NSOP$_{1}$ theories.

## Contribution

It proves transitivity of Kim-independence in NSOP$_{1}$ theories and characterizes Kim-dividing witnesses, advancing understanding of independence in these theories.

## Key findings

- Kim-independence satisfies transitivity in NSOP$_{1}$ theories.
- Witnesses to Kim-dividing are exactly the $ ext{ind}^K$-Morley sequences.
- Several open questions on transitivity and Morley sequences are answered.

## Abstract

We prove several results on the behavior of Kim-independence upon changing the base in NSOP$_{1}$ theories. As a consequence, we prove that Kim-independence satisfies transitivity and that this characterizes NSOP$_{1}$. Moreover, we characterize witnesses to Kim-dividing as exactly the $\ind^{K}$-Morley sequences. We give several applications, answering a number of open questions concerning transitivity, Morley sequences, and local character in NSOP$_{1}$ theories.