Michael Lieberman

Category-theoretic aspects of abstract elementary classes

Abstract We highlight connections between accessible categories and abstract elementary classes (AECs), and provide a dictionary for translating properties and results between the two contexts. We also illustrate a few applications of purely category-theoretic methods to the study of AECs, with model-theoretically novel results. In particular, the category-theoretic approach yields two surprising consequences: a structure theorem for categorical AECs, and a partial stability spectrum for weakly tame AECs.

A topology for galois types in abstract elementary classes

We present a way of topologizing sets of Galois types over structures in abstract elementary classes with amalgamation. In the elementary case, the topologies thus produced refine the syntactic topologies familiar from first order logic. We exhibit a number of natural correspondences between the model-theoretic properties of classes and their constituent models and the topological properties of the associated spaces. Tameness of Galois types, in particular, emerges as a topological separation principle.

Classification theory for accessible categories

We show that a number of results on abstract elementary classes (AECs) hold in accessible categories with concrete directed colimits. In particular, we prove a generalization of a recent result of Boney on tameness under a large cardinal assumption. We also show that such categories support a robust version of the Ehrenfeucht-Mostowski construction. This analysis has the added benefit of producing a purely language-free characterization of AECs, and highlights the precise role played by the coherence axiom.

METRIC ABSTRACT ELEMENTARY CLASSES AS ACCESSIBLE CATEGORIES

We show that metric abstract elementary classes (mAECs) are, in the sense of [LR] (i.e. arXiv:1404.2528), coherent accessible categories with directed colimits, with concrete

μ-Abstract elementary classes and other generalizations ☆

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Universal abstract elementary classes and locally multipresentable categories

-directed colimits and concrete monomorphisms. More broadly, we define a notion of

Topological and category-theoretic aspects of abstract elementary classes

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A Topology for Galois Types in AECs

-concrete AEC---an AEC-like category in which only the

A category-theoretic characterization of almost measurable cardinals

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Tameness in generalized metric structures.

-directed colimits need be concrete---and develop the theory of such categories, beginning with a category-theoretic analogue of Shelahs Presentation Theorem and a proof of the existence of an Ehrenfeucht-Mostowski functor in case the category is large. For mAECs in particular, arguments refining those in [LR] yield a proof that any categorical mAEC is