Stability in Model Theory and Category Theory

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Model theory studies mathematical structures (models) which can be characterised by first-order logical axioms (theories). A number of important mathematical concepts, however, cannot be discussed within the confines of first-order logic, so researchers have been increasingly interested to extend successful techniques of model theory to more general frameworks. One such framework is continuous logic, where we measure the truth of a statement by some value between 0 and 1 rather than just classifying it as "true" or "false". Another example is positive logic, which does not have logical negation built in and it subsumes continuous logic. Finally, there are accessible categories, where one studies categories reminiscent of the category of models of a logical theory by methods of category theory. Stability theory, founded by Shelah in 1970s, has been one of the deepest parts of model theory for decades, culminating in Hrushovski's celebrated model-theoretic proofs of the number-theoretic Manin-Mumford and Mordell-Lang conjectures in the mid-1990s. Hence the desire to generalise stability theory to positive logic and the context of accessible categories that encompass a much wider class of fundamental mathematical examples is perfectly natural. The work of Kim-Pillay from 1990s has shown that stability theory can be studied through various independence relations that tell us which parts of a given structure are related, and which are not. Recently researchers realised that it is possible to use independence relations even in positive logic and accessible categories, and certain parts of stability theory have been generalised to those contexts. Our main goal is to generalise the study of independence relations to the widest possible class of accessible categories including the simple ones, and a certain class of interest in the recent neo-stability theory. At the same time, we will shed light on the stable forking conjecture from the 1990s, by proving a categorical version of it. Adopting a slightly different approach, categorical logic studies models of geometric theories (which includes positive theories) in arbitrary universes called toposes. Because of that, given a geometric theory, one can construct its classifying topos, whose `points' correspond to models, and which affords a certain universal model of the theory. We will study certain positive theories and even some accessible categories as geometric theories through methods of categorical logic, and the exploration of stability in that context will open novel directions for future research. We will strive to enhance the exchange of ideas between model theory and categorical logic, and to build bridges between the two very strong communities in the UK and internationally.