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METRIC ABSTRACT ELEMENTARY CLASSES AS ACCESSIBLE CATEGORIES

Published online by Cambridge University Press:  08 May 2017

M. LIEBERMAN  and
J. ROSICKÝ
M. LIEBERMAN
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS FACULTY OF SCIENCES, MASARYK UNIVERSITY KOTLÁŘSKÁ 2, 611 37 BRNO, CZECH REPUBLIC E-mail: lieberman@math.muni.cz
J. ROSICKÝ
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS FACULTY OF SCIENCES, MASARYK UNIVERSITY KOTLÁŘSKÁ 2, 611 37 BRNO, CZECH REPUBLIC E-mail: rosicky@math.muni.cz
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Abstract

We show that metric abstract elementary classes (mAECs) are, in the sense of [15], coherent accessible categories with directed colimits, with concrete ℵ1-directed colimits and concrete monomorphisms. More broadly, we define a notion of κ-concrete AEC—an AEC-like category in which only the κ-directed colimits need be concrete—and develop the theory of such categories, beginning with a category-theoretic analogue of Shelah’s Presentation Theorem and a proof of the existence of an Ehrenfeucht–Mostowski functor in case the category is large. For mAECs in particular, arguments refining those in [15] yield a proof that any categorical mAEC is μ- d -stable in many cardinals below the categoricity cardinal.

Keywords

03C9503C4803C45abstract model theorymetric abstract elementary classaccessible categoryGalois typesmetric stability

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 3 , September 2017 , pp. 1022 - 1040
Copyright
Copyright © The Association for Symbolic Logic 2017 

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