# On Kim-Independence

## TL;DR

This paper introduces Kim-independence in NSOP$_{1}$ theories, generalizing non-forking independence, and demonstrates its key properties, characterizing NSOP$_{1}$ theories and connecting to known independence notions in specific algebraic structures.

## Contribution

It defines Kim-independence for NSOP$_{1}$ theories and proves it satisfies essential properties, providing a new framework for understanding independence in these theories.

## Key findings

- Kim-independence satisfies Kim's lemma, local character, symmetry, and an independence theorem.
- These properties characterize NSOP$_{1}$ theories.
- Kim-independence aligns with known notions in Frobenius fields and vector spaces with a generic bilinear form.

## Abstract

We study NSOP$_{1}$ theories. We define Kim-independence, which generalizes non-forking independence in simple theories and corresponds to non-forking at a generic scale. We show that Kim-independence satisfies a version of Kim's lemma, local character, symmetry, and an independence theorem and that, moreover, these properties individually characterize NSOP$_{1}$ theories. We describe Kim-independence in several concrete theories and observe that it corresponds to previously studied notions of independence in Frobenius fields and vector spaces with a generic bilinear form.