标题： 有限对齐和非有限对齐高阶图的局部处理以及路径群胚 显示英文标题

标题： A local treatment of finite alignment and path groupoids of nonfinitely aligned higher-rank graphs

摘要： 我们对有限对齐进行局部处理，通过识别任何（不一定是有限对齐的）高阶图的有限对齐部分。 我们证明有限对齐部分本身是一个星系，并与原始高阶图一起形成一个有限对齐的路径相对范畴。 我们证明有限对齐部分的元素正是那些其圆柱集是紧致的元素，这使我们能够为非有限对齐的高阶图提供局部紧致路径和边界路径空间的新定义。 我们将Renault和Williams开发的半群作用及其相关的半直积群胚进行扩展，以定义丰富的Hausdorff路径和边界路径群胚。 根据Renault和Williams的一个结果，对于非有限对齐的$k$-图，这些群胚是可解的。 在有限对齐的情况下，路径群胚与Spielberg的群胚一致，而边界路径群胚则通过Ortega和Pardo的一个结果具有逆半群模型。 显示英文摘要

摘要： We give a local treatment of finite alignment by identifying the finitely aligned part of any (not necessarily finitely aligned) higher-rank graph. We show the finitely aligned part is itself a constellation and forms a finitely aligned relative category of paths together with the original higher-rank graph. We show that the elements of the finitely aligned part are precisely those whose cylinder sets are compact, which allows us to give novel definitions of locally compact path and boundary-path spaces for nonfinitely aligned higher-rank graphs. We extend a semigroup action and the associated semidirect product groupoid developed by Renault and Williams to define ample Hausdorff path and boundary-path groupoids. The groupoids are amenable for nonfinitely aligned $k$-graphs by a result of Renault and Williams. In the finitely aligned case, the path groupoids coincide with Spielberg's groupoids, and the boundary-path groupoid has an inverse semigroup model via a result of Ortega and Pardo.