同伦Hopf-Galois扩张：基础与实例
Homotopic Hopf-Galois extensions: foundations and examples
摘要：
环的Hopf-Galois扩张推广了Galois扩张，用霍普夫代数的余作用代替群的作用。关于群$G$的伽罗瓦扩张是关于$G$的群代数的对偶的Hopf-Galois扩张。Rognes最近定义了一个类似的概念的霍普夫伽罗瓦扩展的范畴内的结构环谱，动机的基本例子单位映射从球谱到$MU$。本文引入了具有相容模型范畴结构的幺半范畴中的同伦Hopf-Galois扩张理论，推广了结构环谱的情形.特别是，我们提供了明确的例子同伦的霍普夫-伽罗瓦扩展在各种类别的兴趣拓扑学家，显示，例如，一个主要的纤维化的单纯幺半群是同伦的霍普夫-伽罗瓦扩展的范畴内的单纯集。我们还研究了同伦Hopf-Galois扩张与下降的关系。
英文摘要：
Hopf-Galois extensions of rings generalize Galois extensions, with the coaction of a Hopf algebra replacing the action of a group. Galois extensions with respect to a group $G$ are the Hopf-Galois extensions with respect to the dual of the group algebra of $G$. Rognes recently defined an analogous notion of Hopf-Galois extensions in the category of structured ring spectra, motivated by the fundamental example of the unit map from the sphere spectrum to $MU$. This article introduces a theory of homotopic Hopf-Galois extensions in a monoidal category with compatible model category structure that generalizes the case of structured ring spectra. In particular, we provide explicit examples of homotopic Hopf-Galois extensions in various categories of interest to topologists, showing that, for example, a principal fibration of simplicial monoids is a homotopic Hopf-Galois extension in the category of simplicial sets. We also investigate the relation of homotopic Hopf-Galois extensions to descent.
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