Dimensions of Ordinals: Set Theory, Homology Theory, and the First Omega Alephs

J. Bergfalk
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引用次数: 3

Abstract

Abstract We describe an organizing framework for the study of infinitary combinatorics. This framework is Čech cohomology. It describes ZFC principles distinguishing among the ordinals of the form $\omega _n$ . More precisely, this framework correlates each $\omega _n$ with an $(n+1)$ -dimensional generalization of Todorcevic’s walks technique, and begins to account for that technique’s “unreasonable effectiveness” on $\omega _1$ . We show in contrast that on higher cardinals $\kappa $ , the existence of these principles is frequently independent of the ZFC axioms. Finally, we detail implications of these phenomena for the computation of strong homology groups and higher derived limits, deriving independence results in algebraic topology and homological algebra, respectively, in the process. Abstract prepared by Jeffrey Bergfalk. E-mail: jeffrey.bergfalk@univie.ac.at
序数的维数:集合论、同调论和第一欧米伽阿莱夫
摘要:我们描述了一个研究无穷组合的组织框架。这个框架是Čech上同调的。它描述了区分形式为$\omega _n$的序数的ZFC原则。更准确地说,这个框架将每个$\omega _n$与托多切维奇的行走技术的$(n+1)$维概括联系起来,并开始解释该技术在$\omega _1$上的“不合理的有效性”。相反,我们表明,在更高的基数$\kappa $上,这些原则的存在往往独立于ZFC公理。最后,我们详细介绍了这些现象对强同调群和高派生极限计算的意义,并在此过程中分别推导出代数拓扑和同调代数中的独立结果。摘要由Jeffrey Bergfalk准备。电子邮件:jeffrey.bergfalk@univie.ac.at
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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