Title:Model Theory in Abstract Elementary Classes

Abstract

stability theory for first order logic developed very quickly after 1970, prompted by the work of Morley, Lachlan, Baldwin, Shelah and Lascar, among others. Earlier on, model theory of infinitary logics, generalized quantifiers had been started (Keisler, Mostowski) but its stability theory was for a long time postponed. Around 25 years ago, a synthesis of many of these logics started, with semantic rather than syntactic emphasis, with focus on the classes of models rather than on the logics axiomatizing them. This synthesis, started by Shelah, is Abstract Elementary Classes. What followed was the development of a very rich and structural stability theory for these classes. Recently, the emphasis has gone back to the interplay with many logics, on the one hand, and with category theory, on the other hand.

I will give four sessions, organized as follows:

1. The Basics: definitions of AECs, examples, the Presentation Theorem. Galois types, stability, independence notions (splitting), the categoricity conjecture and partial results.
2. A proof of categoricity transfer: this part of the minicourse will have a sketch of a proof of categoricity transfer (Vaughtian pairs, rooted types).
3. Categoricity and Large Cardinals: More independence notions (connected to splitting, forking), tameness and large cardinals. The Categoricity Conjecture (consistency).
4. Category theoretic versions: This will be an exploration of recent category theoretic constructions connected with the model theory of AECs. This will also be a summary of the previous lectures, done through the adaptation to category theory.



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