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Internal sizes in \(\mu\)-abstract elementary classes. (English) Zbl 1468.03041

Summary: Working in the context of \(\mu\)-abstract elementary classes (\(\mu\)-AECs) – or, equivalently, accessible categories with all morphisms monomorphisms – we examine the two natural notions of size that occur, namely cardinality of underlying sets and internal size. The latter, purely category-theoretic, notion generalizes e.g. density character in complete metric spaces and cardinality of orthogonal bases in Hilbert spaces. We consider the relationship between these notions under mild set-theoretic hypotheses, including weakenings of the singular cardinal hypothesis. We also establish preliminary results on the existence and categoricity spectra of \(\mu\)-AECs, including specific examples showing dramatic failures of the eventual categoricity conjecture (with categoricity defined using cardinality) in \(\mu\)-AECs.

MSC:

03C48 Abstract elementary classes and related topics
18C35 Accessible and locally presentable categories
03C45 Classification theory, stability, and related concepts in model theory
03C52 Properties of classes of models
03C35 Categoricity and completeness of theories
03E04 Ordered sets and their cofinalities; pcf theory
03E55 Large cardinals
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