Fibrations and Yoneda's lemma in an ∞-cosmos
Fibrations and Yoneda's lemma in an ∞-cosmos
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无穷大宇宙中的纤维振动和米田引理
DOI：
10.1016/j.jpaa.2016.07.003
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发表时间：
2017
影响因子：
0.8
通讯作者：
Dominic R. Verity
中科院分区：
文献类型：
--
作者：
E. Riehl;Dominic R. Verity
我们用∞-范畴和∞-函子来表示∞-宇宙中的对象和态射：一个满足几个公理的简单充实范畴，让人想起一个由不确定对象组成的充实范畴。拟范畴、Segal范畴、完备Segal空间、标记单纯集、迭代完备Segal空间、θ n-空间以及它们各自的纤维形式都是在这个意义上的∞-范畴。本系列以前的工作表明，关于∞-范畴和∞-函子的基本范畴论只能参照∞-宇宙的公理来发展;实际上，大部分工作是同伦2-范畴的内部工作，这是一个严格的2-范畴，包含了∞-范畴、∞-函子和自然变换。在准范畴的∞-宇宙中，我们精确地重新获得了由Joyal和Lurie发展的同一个范畴理论，尽管我们的定义在本质上是2-范畴的，没有利用区分每个模型的组合细节。本文介绍了一类∞-函子--carnanofibration及其群胚变体。笛卡尔纤维化形成了一个基石，在抽象处理的“类范畴”结构的一个洛杉矶街，并发挥了重要作用，在卢里的工作准范畴。在建立了它们的基本理论之后，我们陈述并证明了Yoneda引理，它具有可表示纤维化映射的拟范畴与纤维在其表示元上的拟范畴之间的等价形式。一个配套文件将应用这些结果建立一个演算的模之间的∞范畴，这将被用来定义和研究点态Kan扩张沿着∞函子。
We use the terms∞-categories and∞-functors to mean the objects and morphisms in an∞-cosmos: a simplicially enriched category satisfying a few axioms, reminiscent of an enriched category of fibrant objects. Quasi-categories, Segal categories, complete Segal spaces, marked simplicial sets, iterated complete Segal spaces, θ n-spaces, and fibered versions of each of these are all∞-categories in this sense. Previous work in this series shows that the basic category theory of∞-categories and∞-functors can be developed only in reference to the axioms of an∞-cosmos; indeed, most of the work is internal to the homotopy 2-category, a strict 2-category of∞-categories,∞-functors, and natural transformations. In the∞-cosmos of quasi-categories, we recapture precisely the same category theory developed by Joyal and Lurie, although our definitions are 2-categorical in natural, making no use of the combinatorial details that differentiate each model. In this paper, we introduce cartesian fibrations, a certain class of∞-functors, and their groupoidal variants. Cartesian fibrations form a cornerstone in the abstract treatment of “category-like” structures a la Street and play an important role in Lurie's work on quasi-categories. After setting up their basic theory, we state and prove the Yoneda lemma, which has the form of an equivalence between the quasi-category of maps out of a representable fibration and the quasi-category underlying the fiber over its representing element. A companion paper will apply these results to establish a calculus of modules between∞-categories, which will be used to define and study pointwise Kan extensions along∞-functors.