Kim-forking for hyperimaginaries in NSOP1 theories

Abstract

We adapt the properties of Kim-independence in NSOP1 theories with existence proven in [5],[4] and [2] by Ramsey, Kaplan, Chernikov, Dobrowolski and Kim to hyperimaginaries by adding the assumption of existence for hyperimaginaries. We show that Kim-independence over hyperimaginaries satisfies a version of Kim's lemma, symmetry, the independence theorem, transitivity and witnessing. As applications we adapt Kim's results around colinearity and weak canonical bases from [8] to hyperimaginaries and give some new results about Lascar strong types and Kim-forking using boundedly closed hyperimaginaries.