标题： 2-投影对象理论（附扩展证明） 显示英文标题

标题： A theory of 2-pro-objects (with expanded proofs)

摘要： 格罗滕迪克在范畴$\mathsf{C}$上发展了投影对象的理论。 范畴$\mathsf{Pro}(\mathsf{C})$的基本性质是存在一个嵌入$\mathsf{C} \overset{c}{\longrightarrow} \mathsf{Pro}(\mathsf{C})$，范畴$\mathsf{Pro}(\mathsf{C})$在小的余滤极限下闭合，而且这些极限是自由的，即对于任何在小的余滤极限下闭合的范畴$\mathsf{E}$，与$c$的预复合确定一个范畴等价$\mathcal{C}at(\mathsf{Pro}(\mathsf{C}),\,\mathsf{E})_+ \simeq \mathcal{C}at(\mathsf{C},\, \mathsf{E})$，(其中“$+$”表示保持余滤极限的函子的全子范畴)。 在本文中我们发展了原始对象的二维理论。 给定一个2-范畴$\mathcal{C}$，我们定义一个2-范畴$2\hbox{-}\mathcal{P}ro(\mathcal{C})$，我们称其对象为2-余对象。 我们证明了$2\hbox{-}\mathcal{P}ro(\mathcal{C})$在2-范畴设置中具有所有预期的基本性质，包括对应于上述描述的普遍性质。 我们已经掌握了$\mathcal{C}at$-丰富范畴理论的结果，但我们的理论超越了$\mathcal{C}at$-丰富的情况，因为我们考虑了非严格的概念——伪极限，这通常是实际感兴趣的。 显示英文摘要

摘要： Grothendieck develops the theory of pro-objects over a category $\mathsf{C}$. The fundamental property of the category $\mathsf{Pro}(\mathsf{C})$ is that there is an embedding $\mathsf{C} \overset{c}{\longrightarrow} \mathsf{Pro}(\mathsf{C})$, the category $\mathsf{Pro}(\mathsf{C})$ is closed under small cofiltered limits, and these limits are free in the sense that for any category $\mathsf{E}$ closed under small cofiltered limits, pre-composition with $c$ determines an equivalence of categories $\mathcal{C}at(\mathsf{Pro}(\mathsf{C}),\,\mathsf{E})_+ \simeq \mathcal{C}at(\mathsf{C},\, \mathsf{E})$, (where the "$+$" indicates the full subcategory of the functors preserving cofiltered limits). In this paper we develop a 2-dimensional theory of pro-objects. Given a 2-category $\mathcal{C}$, we define the 2-category $2\hbox{-}\mathcal{P}ro(\mathcal{C})$ 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 universal property corresponding to the one described above. We have at hand the results of $\mathcal{C}at$-enriched category theory, but our theory goes beyond the $\mathcal{C}at$-enriched case since we consider the non strict notion of pseudo-limit, which is usually that of practical interest.