参数化 $\infty$ 类别的局部-全局原理

Hadrian Heine
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引用次数: 0

摘要

我们证明了在任意基$$infty$-类别$\mathcal{C}$上的$infty$-类别的局部-全局原则:我们证明了在任意基$$infty$-类别$\mathcal{B}$上的任意$infty$-类别$\mathcal{C\到 $\mathcal{C}$ 上的\mathcal{C}$ 是由以下数据决定的:对于 $X$ 而言,穿过 $\mathcal{C}$ 对象的等价类集合的纤维集合 $\mathcal{B}_X$ 被赋予了纤维上的自变形空间 $\mathrm{Aut}_X(\mathcal{B})$ 的作用,局部数据,以及局部笛卡尔纤维 $\mathcal{D}\到 $\mathcal{C}$ 和 $\mathrm{Aut}_X(\mathcal{B})$ 线性等价$\mathcal{D}_X \simeq \mathcal{P}(\mathcal{B}_X)$ 到 $\mathcal{B}_X$ 上的预波的$\infty$-类别,即胶合数据。作为应用,我们用左纤维的$infty$类别描述了$[1]$上的小($infty$)$infty$类别,并证明了$\infty$-的内部hom的映射空间的终结式。小$[1]$上的小($infty$)类的内部同和任意小$infty$类$\mathcal{C}上的小($infty$)类的有条件存在的内部同的映射空间的终结式。考虑到$\mathcal{C}$中的函数性,我们可以得到一个推论,即对应的双$infty$类$\mathrm{CORR}$是可呈现的$infty$类的双$infty$类$\mathrm{PR}^L$沿着函数$\infty\mathrm{Cat}的回拉。\来取 presheaves。我们推导出在任何$\infty$-类别$\mathcal{C}$之上的$\infty$-类别都是由正常的涣散 2 函数分类的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A local-global principle for parametrized $\infty$-categories
We prove a local-global principle for $\infty$-categories over any base $\infty$-category $\mathcal{C}$: we show that any $\infty$-category $\mathcal{B} \to \mathcal{C}$ over $\mathcal{C}$ is determined by the following data: the collection of fibers $\mathcal{B}_X$ for $X$ running through the set of equivalence classes of objects of $\mathcal{C}$ endowed with the action of the space of automorphisms $\mathrm{Aut}_X(\mathcal{B})$ on the fiber, the local data, together with a locally cartesian fibration $\mathcal{D} \to \mathcal{C}$ and $\mathrm{Aut}_X(\mathcal{B})$-linear equivalences $\mathcal{D}_X \simeq \mathcal{P}(\mathcal{B}_X)$ to the $\infty$-category of presheaves on $\mathcal{B}_X$, the gluing data. As applications we describe the $\infty$-category of small $\infty$-categories over $[1]$ in terms of the $\infty$-category of left fibrations and prove an end formula for mapping spaces of the internal hom of the $\infty$-category of small $\infty$-categories over $[1]$ and the conditionally existing internal hom of the $\infty$-category of small $\infty$-categories over any small $\infty$-category $\mathcal{C}.$ Considering functoriality in $\mathcal{C}$ we obtain as a corollary that the double $\infty$-category $\mathrm{CORR}$ of correspondences is the pullback of the double $\infty$-category $\mathrm{PR}^L$ of presentable $\infty$-categories along the functor $\infty\mathrm{Cat} \to \mathrm{Pr}^L$ taking presheaves. We deduce that $\infty$-categories over any $\infty$-category $\mathcal{C}$ are classified by normal lax 2-functors.
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