Internal sizes in $\mu$-abstract elementary classes

TL;DR

This paper explores two notions of size in $$-abstract elementary classes, examining their relationship, and investigates categoricity spectra, revealing failures of the eventual categoricity conjecture in this context.

Contribution

It introduces and compares the notions of cardinality and internal size in $$-AECs, and provides initial results on their categoricity spectra and related failures.

Findings

Internal size generalizes density character and basis cardinality.

Relationships between size notions depend on set-theoretic assumptions.

Examples show failures of the eventual categoricity conjecture in $$-AECs.

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.