The 2-category theory of quasi-categories
The 2-category theory of quasi-categories
复制标题
拟范畴的二范畴理论
DOI：
10.1016/j.aim.2015.04.021
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Dominic R. Verity
中科院分区：
文献类型：
--
作者：
E. Riehl;Dominic R. Verity
本文利用二范畴理论重新发展了拟范畴（又称∞范畴）的范畴论基础。我们证明了Joyal的准范畴严格2范畴承认某些弱2极限，其中包括弱逗号对象。我们使用这些逗号拟范畴来编码与极限、边界和附加有关的全称性质，并证明与这些概念有关的预期定理。这些普遍性质有另一种形式，即2类中的绝对提升图，我们展示的是由某些初始顶点或终端顶点的存在点决定的，允许容易地产生示例。这里所介绍的拟直言概念都是等价的，但我们的证明是独立的，更“形式化”。特别地，这些结果立即推广到丰富于准范畴之上的模型范畴。
In this paper we re-develop the foundations of the category theory of quasi-categories (also called ∞-categories) using 2-category theory. We show that Joyal's strict 2-category of quasi-categories admits certain weak 2-limits, among them weak comma objects. We use these comma quasi-categories to encode universal properties relevant to limits, colimits, and adjunctions and prove the expected theorems relating these notions. These universal properties have an alternate form as absolute lifting diagrams in the 2-category, which we show are determined pointwise by the existence of certain initial or terminal vertices, allowing for the easy production of examples.All the quasi-categorical notions introduced here are equivalent to the established ones but our proofs are independent and more “formal”. In particular, these results generalise immediately to model categories enriched over quasi-categories.