粗略类别I：基础
Coarse categories I: foundations
摘要：
根据Roe和其他人（见，例如，[MR1451755]），我们（重新）开发粗糙的几何形状的基础上，采取分类的观点。在本文中，我们集中在离散的情况下，拓扑结构不起作用。我们的理论特别适合于_Roe（C *-）代数_C *（X）及其K-理论在解析方面的发展;我们也希望它能在控制拓扑和代数的严格几何/代数设置中使用。我们把这些问题留给未来的论文。对我们的方法至关重要的是非单位粗空间，以及我们称之为_localized proper_maps（实际上在[MR 1988817]中是隐式的）。我们的_coarse范畴_Crs推广了通常的范畴：它的对象是非酉粗空间，它的态射（局部真）是模_closeness_的粗映射。crs比通常的单位粗范畴丰富得多。因此，它具有所有非零极限和所有余极限。我们研究了其他各种类别的问题。例如，在一个示例中，crs没有终对象，所以我们用一个_termination functor_代替，这在指数对象的开发中将是很重要的（即，"函数空间"），也导致了一个概念的商粗糙空间。为了将我们的方法与标准方法联系起来，我们还研究了Crs与Roe的通常粗范畴之间的关系。最后，我们简要讨论了一些基本的例子和应用。主题包括_度量粗糙空间_，_连续控制_[MR 1277522]，度量和连续控制_粗糙单形_，_sigma-coarse空间_[MR 2225040]，以及商粗糙空间和Roe代数的K-理论之间的关系（对连续控制粗糙空间特别感兴趣）。
英文摘要：
Following Roe and others (see, e.g., [MR1451755]), we (re)develop coarse geometry from the foundations, taking a categorical point of view. In this paper, we concentrate on the discrete case in which topology plays no role. Our theory is particularly suited to the development of the_Roe (C*-)algebras_ C*(X) and their K-theory on the analytic side; we also hope that it will be of use in the strictly geometric/algebraic setting of controlled topology and algebra. We leave these topics to future papers. Crucial to our approach are nonunital coarse spaces, and what we call _locally proper_ maps (which are actually implicit in [MR1988817]). Our_coarse category_ Crs generalizes the usual one: its objects are nonunital coarse spaces and its morphisms (locally proper) coarse maps modulo_closeness_. Crs is much richer than the usual unital coarse category. As such, it has all nonzero limits and all colimits. We examine various other categorical issues. E.g., Crs does not have a terminal object, so we substitute a_termination functor_ which will be important in the development of exponential objects (i.e., "function spaces") and also leads to a notion of_quotient coarse spaces_. To connect our methods with the standard methods, we also examine the relationship between Crs and the usual coarse category of Roe. Finally we briefly discuss some basic examples and applications. Topics include_metric coarse spaces_,_continuous control_ [MR1277522], metric and continuously controlled_coarse simplices_,_sigma-coarse spaces_ [MR2225040], and the relation between quotient coarse spaces and the K-theory of Roe algebras (of particular interest for continuously controlled coarse spaces).
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