论文标题
2-Pro-Objects的理论，一种2模型2类别的理论和2-Pro（C）的2模型结构
A theory of 2-pro-objects, a theory of 2-model 2-categories and the 2-model structure for 2-Pro(C)
论文作者
论文摘要
在六十年代，Grothendieck开发了一个类别的Pro-Objects理论。类别$ pro（c）$的基本属性是，有一个嵌入$ c \ stackrel {c} {c} {\ rightarrow} pro（c）$，$ pro（c）$是在小的cofilter限制下关闭的。 $ cat（pro（c），e）_+ \ simeq cat（c，e）$，（$+ $表示保留辅助限制的函数的完整子类别）。 在这项工作中，我们开发了二维亲对象理论。给定一个2类别$ \ MATHCAL {C} $，我们定义了2类$ 2 $ - $ \ MATHCAL {P} RO（\ MATHCAL {C}）$，其对象我们称为2-Pro-objects。我们证明$ 2 \ hbox { - } \ Mathcal {p} ro（\ Mathcal {c}）$具有与2类别设置相关的所有预期基本属性，包括相应的通用属性。我们给出了封闭2模型2类及其基本属性的封闭的定义。我们离开将来的工作，以构建其同型2类。最后，我们证明我们的2类别$ 2 \ hbox { - } \ Mathcal {p} ro（\ Mathcal {c}）$具有封闭的2模型2类别结构，规定$ \ MATHCAL {c} $具有一个。 这项工作的一部分是开发一个概念框架来处理同型理论中的$ \ check {c} $ ece ece ech nerve，尤其是在强形状理论中。 The $\check{C}$ech nerve is indexed by the categories of covers and of hypercovers, with cover refinements as morphisms, which are not filtered categories, but determine 2-filtered 2-categories on which the $\check{C}$ech nerve is also defined, sends 2-cells into homotopies, and determines a 2-pro-object of simplicial sets.通常，必须将$ \ check {c} $ ece ech Nerve视为同型类别中的亲对象，而失去了显式同型中编码的信息。
In the sixties, Grothendieck developed the theory of pro-objects over a category. The fundamental property of the category $Pro(C)$ is that there is an embedding $C \stackrel{c}{\rightarrow} Pro(C)$, $Pro(C)$ is closed under small cofiltered limits, and these are free in the sense that for any category $E$ closed under small cofiltered limits, pre-composition with $c$ determines an equivalence of categories $Cat(Pro(C),E)_+ \simeq Cat(C,E)$, (the $+$ indicates the full subcategory of the functors that preserve cofiltered limits). In this work we develop a 2-dimensional pro-object theory. Given a 2-category $\mathcal{C}$, we define the 2-category $2$-$\mathcal{P}ro(\mathcal{C})$ whose objects we call 2-pro-objects. We prove that $2\hbox{-}\mathcal{P}ro(\mathcal{C})$ has all the expected basic properties adequately relativized to the 2-categorical setting, including the corresponding universal property. We give an adecuate definition of closed 2-model 2-category and demonstrations of its basic properties. We leave for a future work the construction of its homotopy 2-category. Finally, we prove that our 2-category $2\hbox{-}\mathcal{P}ro(\mathcal{C})$ has a closed 2-model 2-category structure provided that $\mathcal{C}$ has one. Part of the motivation of this work was to develop a conceptual framework to handle the $\check{C}$ech nerve in homotopy theory, in particular in strong shape theory. The $\check{C}$ech nerve is indexed by the categories of covers and of hypercovers, with cover refinements as morphisms, which are not filtered categories, but determine 2-filtered 2-categories on which the $\check{C}$ech nerve is also defined, sends 2-cells into homotopies, and determines a 2-pro-object of simplicial sets. Usually, the $\check{C}$ech nerve has to be considered as a pro-object in the homotopy category, loosing the information encoded in the explicit homotopies.
