Title: Internal sizes in $μ$-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.