# Masaryk University

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# Internal sizes in mu-abstract elementary classes

Authors VASEY Sébastien Bernard 2019 Article in Periodical Journal of Pure and Applied Algebra http://people.math.harvard.edu/~sebv/papers/lrv-sizes/lrv-sizes_v4.pdf http://dx.doi.org/10.1016/j.jpaa.2019.02.004 accessible category; internal size; abstract ekementary class Working in the context of $\mu$-abstract elementary classes 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.