Kan extensions and the calculus of modules for $∞$-categories
Kan extensions and the calculus of modules for $∞$-categories
复制标题
Kan 扩展和 $∞$ 类别的模块演算
DOI：
10.2140/agt.2017.17.189
复制
发表时间：
2015
影响因子：
0.7
通讯作者：
Dominic R. Verity
中科院分区：
文献类型：
--
作者：
E. Riehl;Dominic R. Verity
各种各样的$（\infty，1）$-范畴的模型，包括拟范畴、完备Segal空间、Segal范畴和自然标记单纯集，都可以被认为是$\infty$-宇宙的对象。在一般的$\infty$-cosmos中，其对象我们称之为$\infty$-范畴，我们在$\infty$-范畴之间引入模（也称为原函子或对应），具体表现为具有群胚纤维的适当定义的纤维化的跨度。顾名思义，从$A$到$B$的模块是一个$\infty$-范畴，在适当的意义上，它配备了$A$的左动作和$B$的右动作。应用米田引理的纤维化形式，我们开发了一个通用的模块演算，证明了它们自然地组装成一个多范畴的结构，称为虚拟设备，这是已知的是一个强大的设置，在其中发展正式的范畴理论。使用微积分的模块，它是简单的定义和研究逐点Kan扩展，我们涉及的情况下，carnival封闭$\infty$-cosmoi，限制和colimits图值在$\infty$-类别，介绍了在以前的工作。
Various models of $(\infty,1)$-categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an $\infty$-cosmos. In a generic $\infty$-cosmos, whose objects we call $\infty$-categories, we introduce modules (also called profunctors or correspondences) between $\infty$-categories, incarnated as as spans of suitably-defined fibrations with groupoidal fibers. As the name suggests, a module from $A$ to $B$ is an $\infty$-category equipped with a left action of $A$ and a right action of $B$, in a suitable sense. Applying the fibrational form of the Yoneda lemma, we develop a general calculus of modules, proving that they naturally assemble into a multicategory-like structure called a virtual equipment, which is known to be a robust setting in which to develop formal category theory. Using the calculus of modules, it is straightforward to define and study pointwise Kan extensions, which we relate, in the case of cartesian closed $\infty$-cosmoi, to limits and colimits of diagrams valued in an $\infty$-category, as introduced in previous work.