Michael Lieberman - Generalizing abstract model theory, with an eye towards applications

Michael Lieberman: Generalizing abstract model theory, with an eye towards applications

Abstract model theory arose in the 1980s as a response to the limited applicability of classical model theory - specifically, the model theory of first order logic - to countless classes of interesting objects that occur regularly in mathematical practice. It has long been known, for example, that Artinian rings and Banach spaces do not admit first order axiomatizations, and that the first order theory of the complex numbers with exponentiation is completely intractable. Moreover, one often requires strong notions of homomorphism - pure embeddings, say, or homomorphisms with prescribed quotients - that do not admit a first-order characterization. More general logics suffice (infinitary logics and continuous first order logic often appear in this context), but they are unruly and do not admit any uniform syntactic treatment.

Abstract model theory resolves these issues by discarding logic entirely, focusing on the essential, shared structural properties of the categories of models that can be obtained via these more general logics, thereby encompassing vastly more of the classes of objects occurring in everyday mathematics. In particular, a great deal of attention has been paid to Shelah's abstract elementary classes (AECs), and the metric abstract elementary classes (mAECs) of Hirvonen and Hyttinen (and, increasingly, μ-AECs [1]). I will discuss a single uniform characterization of all such doctrines of abstract model theory (compiled from [2] and [3]), obtained through the magic of category theory. In particular, all of the above doctrines can be characterized as pairs K,U : K → Sets where K is an accessible category with all directed colimits, and U is a faithful functor from K to the category of sets whose properties can be tuned to the desired model-theoretic frequency. This gives clean and easy proofs of major results across the disciplines, which heretofore required independent proofs.

Perhaps more interesting is the observation that there is nothing special about Sets, the category of sets, in this context. Replacing it with Met, the category of complete metric spaces and isometric embeddings, i.e. doing abstract model theory over complete metric spaces, we obtain mAECs. Replacing it with the category of directed graphs, we end up with an abstract model theory over Kripke frames. Replacing it with a suitable category of fuzzy sets, we end up with fuzzy abstract model theory...

The work described is joint with Jiří Rosický (Masaryk University).

[1] Boney, Will, Rami Grossberg, Michael Lieberman, and Jiří Rosický. μ-Abstract Elementary Classes and other generalizations. Submitted 2015.
[2] Lieberman, Michael, and Jiří Rosický. Classification theory for accessible categories. To appear in the Journal of Symbolic Logic. arXiv:1404.2528
[3] Lieberman, Michael, and Jiří Rosický. Metric AECs as accessible categories. Submitted 2015. arXiv:1504.02660