Model theory traditionally studies mathematical structures (models) which can be characterised by first-order logical axioms (a theory). However, a number of important mathematical concepts do not fit in this framework, so researchers have been increasingly interested to generalize successful techniques of model theory to other frameworks, such as continuous logic, positive logic and a very general category-theoretic approach. One of the deepest parts of model theory is stability theory, which has inspired a great deal of research with countless applications, even when limited to the traditional first-order framework. Stability theory can be studied through independence relations that tell us which parts of a given structure are related, and which are not. Such independence relations work especially well in the stable, simple and NSOP1 settings, which relate to the complexity of the structures involved. Modern formulations of independence relations allow for a clear translation to categorical language.
We propose to build on pioneering work to generalize stability-theoretic tools for the simple and NSOP1 settings to the categorical framework. This overarching goal is split up in three objectives.
(O1) Develop categorical independence, primarily in the unstable settings.
(O2) Establish novel approaches to the long-standing stable forking conjecture.
(O3) Found an enriched model theory and stability theory through enriched categories.
This ambitious generalization will be achieved by combining my experience in modern model theory and stability theory, with the complementary expertise of my supervisor, John Bourke, and my advisors, who are world-leaders in categorical model theory and other categorical aspects relevant to this project, such as enriched category theory. On an international level this project will build a bridge between the model theory and category theory communities, thus enlarging my network and enhancing my prospects towards an academic career.
