This is the title of two talks given at the CMU model theory seminar on September 21 and 28, 2015. The talks present the corresponding paper.
Abstract elementary classes (AECs) are a purely semantic framework to study the model theory of many different infinitary logics and algebraic classes. Roughly speaking, an AEC is a class of models together with a partial ordering satisfying some of the basic properties of elementary substructure in first-order model theory.
In Chapter IV of his book on AECs, Shelah has shown that an AEC categorical (i.e. with a single model up to isomorphism) in a high-enough cardinal has, in some suitable cardinals, a structural property called amalgamation. A key lemma in his proof is that in an AEC categorical in a high-enough cardinal, the partial ordering ends up being exactly (at least on a tail of the class) the elementary substructure relation of an infinitary logic.
Recently, Will Boney and the speaker identified a gap in Shelah's proof of the key lemma. In this series of talks, we will explain what the gap is and how to fix it. The proof is combinatorial, the main ingredients being model-theoretic forcing and the proper class version of Fodor's lemma. The talks will be self-contained and the required background on AECs will be introduced as we go along. This is joint work with Will Boney.