This is the title of a talk given at the University of Maryland Logic seminar on May 14, 2019.
Morley's categoricity theorem says that a countable first-order theory with a single model in some uncountable cardinal has a single model in all uncountable cardinals. The proof has had numerous consequences, including the development of stability theory, and in particular the invention by Shelah of forking, a far reaching generalization of algebraic independence in fields. Shelah has conjectured that a generalization of Morley's result should hold in any abstract elementary class (AEC). Roughly, an AEC is a category that behaves like the category of models of a first-order theory with elementary embedding. The framework encompasses for example classes of models of logics with infinite conjunctions and disjunctions.
In this talk, I will survey progress on Shelah's eventual categoricity conjecture, including a proof from large cardinals (joint with Shelah) and a full characterization of the categoricity spectrum in AECs with amalgamation. These proofs have already suggested several new directions, including a theory of forking in accessible categories connecting the field with homotopy theory (joint with Lieberman and Rosický).
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.