Sizes and filtrations in accessible categories
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SIZES AND FILTRATIONS IN ACCESSIBLE CATEGORIES
BY
Michael Lieberman Department of Mathematics and Statistics, Faculty of Science Masaryk University, Brno 611 37, Czech Republic and Institute of Mathematics, Faculty of Mechanical Engineering Brno University of Technology, Brno 616 69, Czech Republic e-mail: [email protected] AND
ˇ´ı Rosicky ´∗ Jir Department of Mathematics and Statistics, Faculty of Science Masaryk University, Brno 611 37, Czech Republic e-mail: [email protected] URL: http://www.math.muni.cz/˜rosicky/ AND
Sebastien Vasey Department of Mathematics, Harvard University Cambridge, MA 02138, USA e-mail: [email protected] URL: http://math.harvard.edu/˜sebv/
∗ The second author is supported by the Grant agency of the Czech republic under
the grant 19-00902S. Received March 1, 2019 and in revised form June 5, 2019
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´ AND S. VASEY M. LIEBERMAN, J. ROSICKY
Isr. J. Math.
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 L¨ owenheim–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 cofinality. 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.
1. Introduction Recent years have seen a burst of research activity connecting accessible categories with abstract model theory. Abstract model theory, which has always had the aim of generalizing—in a uniform way—fragments of the rich classification theory of first order logic to encompass the broader nonelementary classes of structures that abound in mathematics proper, is perhaps most closely identified with abstract elementary classes (AECs, [She87]), but also encompasses metric AECs (mAECs, [HH09]), compact abstract theories (cats, [BY05]), and a host of other proposed frameworks. While accessible categories appear in many areas that model theory fears to tread—homotopy theory, for example— they are, fundamentally, generalized categories of models, and the ambition to recover a portion of classification theory in this context has been present since the very beginning, [MP89, p. 6]. That these fields are connected has been evident for some time—the first recognition that AECs are special accessible categories came independently in [BR12] and [Lie11]—but it is only recently that a precise middle-ground has been identified: the μ-AECs of [BGL+ 16]. While we recall the precise definition of μ-AEC below, we note that they are a natural generalization of AECs in which the ambient language is allowed to be μ-ary, one assumes closure only under μ-directed unions rather than unions of arbitrary chains, and the L¨owenheim–Skolem–Tarski property is weakened accordingly. The motivations for this de
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