On Kim independence
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.