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THE KIM–PILLAY THEOREM FOR ABSTRACT ELEMENTARY CATEGORIES

Part of: Model theory Categories and theories

Published online by Cambridge University Press:  30 October 2020

MARK KAMSMA
MARK KAMSMA*
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF EAST ANGLIANORWICH, NORFOLK, NR4 7TJ, UKE-mail: [email protected]: https://markkamsma.nl
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Abstract

We introduce the framework of AECats (abstract elementary categories), generalizing both the category of models of some first-order theory and the category of subsets of models. Any AEC and any compact abstract theory (“cat”, as introduced by Ben-Yaacov) forms an AECat. In particular, we find applications in positive logic and continuous logic: the category of (subsets of) models of a positive or continuous theory is an AECat. The Kim–Pillay theorem for first-order logic characterizes simple theories by the properties dividing independence has. We prove a version of the Kim–Pillay theorem for AECats with the amalgamation property, generalizing the first-order version and existing versions for positive logic.

Keywords

dividingaccessible categorysimple theoryabstract elementary classindependence relationabstract elementary category

MSC classification

Primary: 03C45: Classification theory, stability and related concepts
Secondary: 03C48: Abstract elementary classes and related topics 03C52: Properties of classes of models 18C35: Accessible and locally presentable categories
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 4 , December 2020 , pp. 1717 - 1741
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Copyright
© The Association for Symbolic Logic 2020

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