Internal sizes in $\mu$-abstract elementary classes

Abstract

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.

27 pages

References in corpus (4)

• Shelah's eventual categoricity conjecture in universal classes: part I
• $\mu$-Abstract Elementary Classes and other generalizations
• Toward a stability theory of tame abstract elementary classes
• Universal abstract elementary classes and locally multipresentable categories

Cited by in corpus (1)

• All $(\infty ,1)$-toposes have strict univalent universes