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THE KIM–PILLAY THEOREM FOR ABSTRACT ELEMENTARY CATEGORIES

Published online by Cambridge University Press:  30 October 2020

MARK KAMSMA*
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF EAST ANGLIA NORWICH, NORFOLK, NR4 7TJ, UK E-mail: m.kamsma@uea.ac.uk URL: https://markkamsma.nl

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.

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Copyright
© The Association for Symbolic Logic 2020

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THE KIM–PILLAY THEOREM FOR ABSTRACT ELEMENTARY CATEGORIES
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