Algebra Seminar talk
Ivan Di Liberti (Universitas Masarykiana Brunensis)
Topos theoretic approaches to Abstract model theory
Model theory is the study of mathematical structures axiomatizable in classical first order logic; these structures are organized in elementary classes, which are the main object of investigation of model theorists.
In the 1970's Shelah used abstract elementary classes (AECs) as a framework to apply model theoretic techniques to infinitary logics. It is always a relevant challenge to provide syntactic axiomatizations of an abstract elementary class. In full generality this problem leads to complications and the results are not completely satisfactory.
We try to give a categorical approach to this problem via topos theory. In a nutshell, topos theory is where geometry meets logic. A topos is both a generalized theory and a generalized space. Given an AEC, say A, (and even more generally an accessible category with directed colimits) we introduce the Scott topos S(A) of A. Since topoi can be seen as a semantic incarnation of theories, the Scott topos of an AEC is a candidate axiomatization of the AEC itself. We show that this intuition is consistent and fruitful.