Talk given in Prague at the 16th CLMPST, session on some recent development in model theory, August 8, 2019. See the Slides.
The classification theory of elementary classes was started by Michael Morley in the early sixties, when he proved that a countable first-order theory with a single model in some uncountable cardinal has a single model in all uncountable cardinals. The proof of this result, now called Morley's categoricity theorem, led to the development of forking, a notion of independence jointly generalizing linear independence in vector spaces and algebraic independence in fields and now a central pillar of modern model theory.
In recent years, it has become apparent that the theory of forking can also be developed in several non-elementary contexts. Prime among those are the axiomatic frameworks of accessible categories and abstract elementary classes (AECs), encompassing classes of models of any reasonable infinitary logics. A test question to judge progress in this direction is the forty year old eventual categoricity conjecture of Shelah, which says that a version of Morley's categoricity theorem should hold of any AEC. I will survey recent developments, including the connections with category theory and large cardinals, a theory of forking in accessible categories (joint with M. Lieberman and J. Rosický), as well as the resolution of the eventual categoricity conjecture from large cardinals (joint with S. Shelah).
Sebastien Vasey, Accessible categories, set theory, and model theory: an invitation, Preprint: pdf arXiv, 67 pages.
Sebastien Vasey, The categoricity spectrum of large abstract elementary classes, preprint: pdf arXiv, 46 pages.
Saharon Shelah and Sebastien Vasey, Categoricity and multidimensional diagrams, preprint: pdf arXiv, 63 pages.
Michael Lieberman, Jiří Rosický, and Sebastien Vasey, Weak factorization systems and stable independence, Preprint: pdf arXiv.
Michael Lieberman, Jiří Rosický, and Sebastien Vasey, Forking independence from the categorical point of view, Advances in Mathematics 346 (2019), 719–772. Publisher version pdf arXiv.