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 categorytheoretic, 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 settheoretic 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2017
 arXiv:
 arXiv:1708.06782
 Bibcode:
 2017arXiv170806782L
 Keywords:

 Mathematics  Logic;
 Mathematics  Category Theory;
 03C48 (Primary);
 18C35;
 03C45;
 03C52;
 03C55;
 03C75;
 03E04;
 03E5518C35;
 03C52;
 03C55;
 03C75 (Secondary)
 EPrint:
 27 pages