Internal sizes in $��$-abstract elementary classes

Working in the context of $��$-abstract elementary classes ($��$-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 $��$-AECs, including specific examples showing dramatic failures of the eventual categoricity conjecture (with categoricity defined using cardinality) in $��$-AECs.
27 pages