Independence Relations in Abstract Elementary Categories

Abstract In model theory, a branch of mathematical logic, we can classify structures based on their logical complexity. This yields the so-called stability hierarchy. Independence relations play an important role in this An independence relation tells us which subsets structure contain information about each other, for example, linear vector spaces such relation. Some classes hierarchy are stable, simple, and NSOP $_1$ , being contained next. For these there exists Kim-Pillay style theorem. Such theorem describes interaction between simplicity is equivalent to admitting certain relation, must then be unique. All above classically takes place full first-order logic. Parts it have already been generalised other frameworks, as continuous positive even very general category-theoretic framework. thesis continue work. We introduce framework AECats, specific kind accessible category. prove that at most one or -like AECat. thus recover (part of) original notions long dividing, isi-dividing, Kim-dividing, classical dividing Kim-dividing but they work well without compactness. Switching generalise theories over existentially closed models has all nice properties known also provide theorem: theory if only enough given by Kim-dividing. prepared Mark Kamsma. E-mail: [email protected] . URL: https://markkamsma.nl/phd-thesis