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CELLULAR CATEGORIES AND STABLE INDEPENDENCE

Part of: General and miscellaneous Applied homological algebra and category theory Model theory Categories and theories

Published online by Cambridge University Press:  18 May 2022

MICHAEL LIEBERMAN Open the ORCID record for MICHAEL LIEBERMAN [Opens in a new window] ,
JIŘÍ ROSICKÝ Open the ORCID record for JIŘÍ ROSICKÝ [Opens in a new window]  and
SEBASTIEN VASEY Open the ORCID record for SEBASTIEN VASEY [Opens in a new window]
MICHAEL LIEBERMAN*
Affiliation:
INSTITUTE OF MATHEMATICS, FACULTY OF MECHANICAL ENGINEERING BRNO UNIVERSITY OF TECHNOLOGY BRNO, CZECH REPUBLIC URL: https://math.fme.vutbr.cz/Home/lieberman
JIŘÍ ROSICKÝ
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS FACULTY OF SCIENCE MASARYK UNIVERSITY BRNO, CZECH REPUBLIC E-mail: rosicky@math.muni.cz URL: http://www.math.muni.cz/~rosicky/
*
E-mail: qmlieberman@vutbr.cz
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Abstract

We exhibit a bridge between the theory of cellular categories, used in algebraic topology and homological algebra, and the model-theoretic notion of stable independence. Roughly speaking, we show that the combinatorial cellular categories (those where, in a precise sense, the cellular morphisms are generated by a set) are exactly those that give rise to stable independence notions. We give two applications: on the one hand, we show that the abstract elementary classes of roots of Ext studied by Baldwin–Eklof–Trlifaj are stable and tame. On the other hand, we give a simpler proof (in a special case) that combinatorial categories are closed under 2-limits, a theorem of Makkai and Rosický.

Keywords

cellular categoriesforkingstable independenceabstract elementary classcofibrantly generatedroots of Ext

MSC classification

Primary: 18C35: Accessible and locally presentable categories
Secondary: 03C45: Classification theory, stability and related concepts 03C48: Abstract elementary classes and related topics 03C52: Properties of classes of models 03C55: Set-theoretic model theory 16B50: Category-theoretic methods and results (except as in ) 55U35: Abstract and axiomatic homotopy theory
Type
Article
Information
The Journal of Symbolic Logic , Volume 88 , Issue 2 , June 2023 , pp. 811 - 834
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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