Topological and Category-Theoretic Aspects of Abstract Elementary Classes.

Abstract

We consider the behavior of Galois types in abstract elementary classes (AECs), and introduce several new techniques for use in the analysis of the associated stability spectra. More broadly, we develop novel perspectives on AECs---topological and category-theoretic---from which these techniques flow, and which hold considerable promise as lines of future investigation.

After a presentation of the preliminaries in Chapter 2, we give a method of topologizing sets of Galois types over structures in AECs with amalgamation. The resulting spaces---analogues of the Stone spaces of syntactic types---support, among other things, natural correspondences between their topological properties and semantic properties of the AEC (tameness, for example, emerges as a separation principle).

In Chapter 4, we note that the newfound topological structure yields a family of Morley-like ranks, along with a new notion of total transcendence. We show that in tame AECs, total transcendence follows from stability in certain cardinals, and that total transcendence, in turn, allows us to bound the number of types over large models. This leads to several upward stability transfer results, one of which generalizes a result of Baldwin, Kueker and VanDieren. The same analysis works in weakly tame AECs provided that they are also weakly stable, a notion that arises in the context of accessible categories.

In Chapter 5, we analyze the category-theoretic structure of AECs, and give an axiomatization of AECs as accessible subcategories of their ambient categories of structures. We also give a dictionary for translating notions from the theory of accessible categories into the language of AECs, and vice versa. Weak stability occurs in any accessible category---hence in any AEC---and, since this is what we require to conclude stability in weakly tame AECs, we get the beginnings of a stability spectrum in this context.

We close with a curious result: an equivalence between the class of large structures in a categorical AEC and a category of sets with actions of the monoid of endomorphisms of the categoricity structure, effectively reducing the AEC to a simple concrete category.