受控理论、范畴化与同伦化
Controlled theories, categorification, and homotopification
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中文总结 AI 辅助
本文引入受控理论概念，用于高阶范畴代数研究。定义其变形概念，构建函子式范畴化与同伦化，得到相关理论。还获得∞ - 群新模型及相干类群\(E_∞\) - 空间模型，为无限环空间建模。
中文摘要 AI 辅助
本文引入受控理论的概念，其最初在作者论文中提出，作为研究高阶范畴代数的结构工具。我们在笛卡尔闭范畴中定义了预层和受控理论的变形概念。此外，证明了受控理论的变形自然产生基于同一基础范畴丰富的Lawvere理论。构建了受控理论的函子式一维范畴化和同伦化，分别得到Lawvere 2 - 理论和单纯集丰富的Lawvere理论。作为应用，获得了∞ - 群的新模型并构建了相干类群\(E_∞\) - 空间的模型，未来工作将表明其为无限环空间建模。
英文摘要
In this paper, we introduce the notion of a controlled theory, originally developed in the author's thesis, as a structural tool for the study of higher categorical algebra. We define a notion of deformation for pros and controlled theories in a cartesian closed category. Furthermore, we show that deformations of controlled theories naturally produce Lawvere theories enriched over the same base category. We construct functorial one-dimensional categorifications and homotopifications of controlled theories, yielding Lawvere $2$-theories and Lawvere theories enriched in simplicial sets, respectively. As an application, we obtain a new model for $\infty$-groups and construct a model of coherent group-like $E_\infty$-spaces, which we will show in future work models infinite loop spaces.
