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Sizes and filtrations in accessible categories

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Abstract

Accessible categories admit a purely category-theoretic replacement for cardinality: the internal size. Generalizing results and methods from [LRV19b], 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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Authors and Affiliations

  1. Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Brno, 611 37, Czech Republic

    Michael Lieberman & Jiří Rosický

  2. Institute of Mathematics, Faculty of Mechanical Engineering, Brno University of Technology, Brno, 616 69, Czech Republic

    Michael Lieberman

  3. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Sebastien Vasey

Authors
  1. Michael Lieberman
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  2. Jiří Rosický
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  3. Sebastien Vasey
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Corresponding author

Correspondence to Sebastien Vasey.

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The second author is supported by the Grant agency of the Czech republic under

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Lieberman, M., Rosický, J. & Vasey, S. Sizes and filtrations in accessible categories. Isr. J. Math. 238, 243–278 (2020). https://doi.org/10.1007/s11856-020-2018-8

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  • DOI: https://doi.org/10.1007/s11856-020-2018-8

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