PhD thesis: Independence Relations in Abstract Elementary Categories
On 1 October 2018 I started my PhD at the University of East Anglia, under supervision of Jonathan Kirby. On 17 September 2021 I submitted my PhD thesis, and I passed my viva on 6 December 2021. The title of my thesis is:
Independence Relations in Abstract Elementary Categories
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Papers based on the thesis
The main results from the thesis also appear in the following papers.
- The Kim-Pillay theorem for Abstract Elementary Categories, Journal of Symbolic Logic, 85(4):1717-1741, December 2020 (journal, arXiv).
- NSOP1-like independence in AECats, Journal of Symbolic Logic, December 2022 (journal, arXiv).
- Kim-independence in positive logic, joint with Jan Dobrowolski, Model Theory, 1(1):55-113, June 2022 (journal, arXiv).
Roughly speaking, chapters 3, 4 and 5 can be found in papers 1 and 2. Chapter 6 is essentially paper 3.
In model theory, a branch of mathematical logic, we can classify mathematical structures based on their logical complexity. This yields the so-called stability hierarchy. Independence relations play an important role in this stability hierarchy. An independence relation tells us which subsets of a structure contain information about each other, for example: linear independence in vector spaces yields such a relation.
Some important classes in the stability hierarchy are stable, simple and NSOP1, each being contained in the next. For each of these classes there exists a so-called Kim-Pillay style theorem. Such a theorem describes the interaction between independence relations and the stability hierarchy. For example: simplicity is equivalent to admitting a certain independence relation, which must then be unique.
All of the above classically takes place in full first-order logic. Parts of it have already been generalised to other frameworks, such as continuous logic, positive logic and even a very general category-theoretic framework. In this thesis we continue this work.
We introduce the framework of AECats, which are a specific kind of accessible category. We prove that there can be at most one stable, simple or NSOP1-like independence relation in an AECat. We thus recover (part of) the original stability hierarchy. For this we introduce the notions of long dividing, isi-dividing and long Kim-dividing, which are based on the classical notions of dividing and Kim-dividing but are such that they work well without compactness.
Switching frameworks, we generalise Kim-dividing in NSOP1 theories to positive logic. We prove that Kim-dividing over existentially closed models has all the nice properties that it is known to have in full first-order logic. We also provide a full Kim-Pillay style theorem: a positive theory is NSOP1 if and only if there is a nice enough independence relation, which then must be given by Kim-dividing.