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arXiv · 1708.06782

Internal sizes in $\mu$-abstract elementary classes

Michael Lieberman·Jiří Rosický·Sebastien Vasey
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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.

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Michael Lieberman, Jiří Rosický, Sebastien Vasey. 2017-08-22. Internal sizes in $\mu$-abstract elementary classes. https://doi.org/10.1016/j.jpaa.2019.02.004

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