CLASSIFYING SPACES AND THE LASCAR GROUP

Author:

CAMPION TIM,COUSINS GREG,YE JINHEORCID

Abstract

AbstractWe 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.

Publisher

Cambridge University Press (CUP)

Subject

Logic,Philosophy

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Cited by 1 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. NON-TRIVIAL HIGHER HOMOTOPY OF FIRST-ORDER THEORIES;The Journal of Symbolic Logic;2024-01-11

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