Pro-categories in homotopy theory
Pro-categories in homotopy theory
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同伦理论中的亲范畴
DOI：
10.2140/agt.2017.17.567
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Geoffroy Horel
中科院分区：
文献类型：
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作者：
Ilan Barnea;Yonatan Harpaz;Geoffroy Horel
本文的目标是证明由伊萨克森（Isaksen）、施兰克（Schlank）以及第一作者所研究的关于投射范畴（pro - categories）的模型范畴方法与卢里（Lurie）所发展的$\infty$-范畴方法之间的等价性。我们描述了主要结果的三个应用。在第一个应用中，我们使用（主要结果的一个对偶版本）来给出关于一个$\omega$-组合模型范畴的充分条件，这些条件确保其底层的$\infty$-范畴是$\omega$-可表示的。在第二个应用中，我们考虑单纯平展层（simplicial etale sheaves）的投射范畴，并利用它来表明任何格罗滕迪克拓扑斯（Grothendieck topos）的拓扑实现与相关$\infty$-拓扑斯的超完备化的形状一致。在第三个应用中，我们表明在有限完备同伦理论（profinite homotopy theory）中出现的几个模型范畴确实是有限完备空间（profinite spaces）的$\infty$-范畴的模型。作为一个副产品，我们得到了这些模型之间新的奎伦等价（Quillen equivalences），并且还得到了一个例子，它否定地解决了拉普蒂斯（Raptis）提出的一个问题。
The goal of this paper is to prove an equivalence between the model categorical approach to pro-categories, as studied by Isaksen, Schlank and the first author, and the $\infty$-categorical approach, as developed by Lurie. Three applications of our main result are described. In the first application we use (a dual version of) our main result to give sufficient conditions on an $\omega$-combinatorial model category, which insure that its underlying $\infty$-category is $\omega$-presentable. In the second application we consider the pro-category of simplicial etale sheaves and use it to show that the topological realization of any Grothendieck topos coincides with the shape of the hyper-completion of the associated $\infty$-topos. In the third application we show that several model categories arising in profinite homotopy theory are indeed models for the $\infty$-category of profinite spaces. As a byproduct we obtain new Quillen equivalences between these models, and also obtain an example which settles negatively a question raised by Raptis.