AI 中文总结
本文为粗空间的粗子空间提供范畴框架，引入受控全关系，证明相关赋值的函子性及偏序集同构，阐述渐近不相交性，为粗子空间框架提供解释，刻画其为范畴内在对象，还为恰当性准则提供粗几何解释基础。
AI 中文摘要
在本文中，我们为理解粗空间的粗子空间提供了一个范畴框架。首先，我们引入粗空间之间受控全关系的概念，并表明其态射为受控全关系的接近类的范畴同构于用受控映射的接近类定义的传统粗空间范畴。接着，我们证明将每个粗空间与其粗子空间的有限并偏序集相关联的赋值是函子性的，并证明这个偏序集自然同构于粗空间范畴中的子对象偏序集。此外，我们阐述了粗子空间之间的渐近不相交性，并表明单态射保持这种关系。这些结果为Leitner - Vigolo [《数学讲义》(2023)]引入的粗子空间框架提供了范畴解释，并将粗子空间刻画为粗空间范畴内在的对象。它们还为Kobayashi [《数学年刊》(1989);《李理论杂志》(1996)]和Benoist [《数学年刊》(1996)]建立的恰当性准则提供了粗几何解释的基础（参见Nagaya - Ogawa - Okuda [《日本学术振兴会会报A辑》(2025)]）。
英文摘要
In this paper, we provide a categorical framework for understanding coarse subspaces of coarse spaces. First, we introduce the notion of a controlled total relation between coarse spaces and show that the category whose morphisms are closeness classes of controlled total relations is isomorphic to the conventional category of coarse spaces defined using closeness classes of controlled maps. Next, we show that the assignment associating to each coarse space the finite-join partially ordered set of its coarse subspaces is functorial, and prove that this partially ordered set is naturally isomorphic to the poset of subobjects in the category of coarse spaces. Furthermore, we formulate asymptotic disjointness between coarse subspaces and show that mono-morphisms preserve this relation. These results provide a categorical interpretation of the framework of coarse subspaces introduced by Leitner--Vigolo [Lecture Notes in Math.~(2023)] and characterize coarse subspaces as objects intrinsic to the category of coarse spaces. They also provide a foundation for a coarse-geometric interpretation of the properness criterion established by Kobayashi [Math.~Ann.~(1989); J.~Lie Theory (1996)] and Benoist [Ann.~of Math.~(1996)] (cf.~Nagaya--Ogawa--Okuda [Proc.~Japan Acad.~Ser.~A (2025)]).
Comments53 pages
