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CLASSIFYING SPACES AND THE LASCAR GROUP

Part of: Applied homological algebra and category theory Model theory

Published online by Cambridge University Press:  13 September 2021

TIM CAMPION ,
GREG COUSINS  and
JINHE YE Open the ORCID record for JINHE YE [Opens in a new window]
TIM CAMPION
Affiliation:
DEPARTMENT OF MATHEMATICS JOHNS HOPKINS UNIVERSITYBALTIMORE, MD, USAE-mail: tcampio1@jh.edu
GREG COUSINS
Affiliation:
DEPARTMENT OF MATHEMATICS & STATISTICS McMASTER UNIVERSITYHAMILTON, ON, CANADAE-mail: cousingd@mcmaster.ca
JINHE YE
Affiliation:
INSTITUT DE MATHÉMATIQUES DE JUSSIEU—PARIS RIVE GAUCHE SORBONNE UNIVERSITÉPARIS, FRANCEE-mail: jinhe.ye@imj-prg.fr
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Abstract

We show that the Lascar group $\operatorname {Gal}_L(T)$ of a first-order theory T is naturally isomorphic to the fundamental group $\pi _1(|\mathrm {Mod}(T)|)$ of the classifying space of the category of models of T and elementary embeddings. We use this identification to compute the Lascar groups of several example theories via homotopy-theoretic methods, and in fact completely characterize the homotopy type of $|\mathrm {Mod}(T)|$ for these theories T. It turns out that in each of these cases, $|\operatorname {Mod}(T)|$ is aspherical, i.e., its higher homotopy groups vanish. This raises the question of which homotopy types are of the form $|\mathrm {Mod}(T)|$ in general. As a preliminary step towards answering this question, we show that every homotopy type is of the form $|\mathcal {C}|$ where $\mathcal {C}$ is an Abstract Elementary Class with amalgamation for $\kappa $ -small objects, where $\kappa $ may be taken arbitrarily large. This result is improved in another paper.

Keywords

classifying spaceLascar grouphomotopy type

MSC classification

Primary: 03C48: Abstract elementary classes and related topics
Secondary: 03C52: Properties of classes of models 55U10: Simplicial sets and complexes
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 4 , December 2021 , pp. 1396 - 1431
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Association for Symbolic Logic

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