在一个无穷拓扑斯中的高阶覆盖空间
Higher covering spaces in an $\infty$-topos
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中文总结 AI 辅助
在无穷拓扑斯中发展\(n\) - 覆盖映射理论，证明\(n\) - 覆盖与\(\Pi_n(X,x)\)无穷作用的\(n\) - 范畴等价性，研究甲板变换\(n\) - 群，给出正规\(n\) - 覆盖分类，方法依赖多方面研究，结果恢复经典并适用于多种无穷拓扑斯。
中文摘要 AI 辅助
我们在一个固定的无穷拓扑斯\(\mathscr{E}\)中，通过与经典覆盖空间类比，发展了一种关于\((n - 1)\) - 截断映射（称为\(n\) - 覆盖映射）的系统理论。我们证明了在一个有指连通对象\((X,x)\)上的\(n\) - 覆盖与基本\(n\) - 群\(\Pi_n(X,x)\)对\((n - 1)\) - 截断对象的无穷作用之间的\(n\) - 范畴等价性，这限制为根据\(\Pi_n(X,x)\)的子\(n\) - 群对有指连通\(n\) - 覆盖进行分类。我们研究了甲板变换的\(n\) - 群\(\mathscr{D}\mathrm{eck}(p)\)，将其与纤维\(F\)的\(\Pi_n(X,x)\) - 等变自同构等同起来。对于正规\(n\) - 覆盖，它进一步被描述为\(\Pi_n(X,x)\)的商，从而根据\(\pi_n(X,x)\)的正规子群对这种覆盖进行分类。对于任意\(n\) - 覆盖，甲板\(n\) - 群作为一个合适正规化子的商出现。我们的方法依赖于对\(n\) - 群及其无穷作用的仔细研究、单值宇宙的使用以及内部约当嵌入。当\(n = 1\)且\(\mathscr{E}\)是同伦类型的无穷范畴时，我们的结果恢复了经典的覆盖空间理论。我们还在层和étale无穷拓扑斯以及凝聚无穷拓扑斯中说明了该理论，在层和étale无穷拓扑斯中外部甲板群恢复了基的上同调，在凝聚无穷拓扑斯中它恢复了流形的\(1\) - 覆盖理论。
英文摘要
We develop a systematic theory of $(n-1)$-truncated maps, called $n$-covering maps, in a fixed $\infty$-topos $\mathscr{E}$, guided by the analogy with classical covering spaces. We prove an equivalence of $n$-categories between $n$-coverings over a pointed connected object $(X,x)$ and $\infty$-actions of the fundamental $n$-group $Π_n(X,x)$ on $(n-1)$-truncated objects, which restricts to a classification of pointed connected $n$-coverings in terms of sub-$n$-groups of $Π_n(X,x)$. We study the $n$-group of deck transformations $\mathscr{D}\mathrm{eck}(p)$, identifying it with $Π_n(X,x)$-equivariant autoequivalences of the fiber $F$. For normal $n$-coverings, it is further described as a quotient of $Π_n(X,x)$, yielding a classification of such coverings in terms of normal subgroups of $π_n(X,x)$. For an arbitrary $n$-covering, the deck $n$-group arises as a quotient of a suitable normalizer. Our approach relies on a careful study of $n$-groups and their $\infty$-actions, on the use of univalent universes, and on an internal Yoneda embedding. When $n=1$ and $\mathscr{E}$ is the $\infty$-category of homotopy types, our results recover the classical theory of covering spaces. We further illustrate the theory in sheaf and étale $\infty$-topoi, where the external deck group recovers cohomology of the base, and in cohesive $\infty$-topoi, where it recovers the $1$-covering theory of manifolds.
