Publication details

Sizes and filtrations in accessible categories

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LIEBERMAN Michael ROSICKÝ Jiří VASEY Sébastien Bernard

Year of publication 2020
Type Article in Periodical
Magazine / Source Israel Journal of Mathematics
MU Faculty or unit

Faculty of Science

Keywords internal size; presentability rank; existence spectrum; accessibility spectrum; filtrations; singular cardinal hypothesis
Description Accessible categories admit a purely category-theoretic replacement for cardinality: the internal size. We examine set-theoretic problems related to internal sizes and prove several Löwenheim–Skolem theorems for accessible categories. For example, assuming the singular cardinal hypothesis, we show that a large accessible category has an object in all internal sizes of high enough co-finality. We also prove that accessible categories with directed colimits have filtrations: any object of sufficiently high internal size is (the retract of) a colimit of a chain of strictly smaller objects.
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