发表机构
Departamento de Matemáticas, Universidade de Santiago de Compostela（圣地亚哥-德孔波斯特拉大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对空间图系统提出拓扑复杂度概念，定义为端点评估映射的截面范畴，通过轨道形状选择实现范畴压缩，并给出上下界，推广了等变情形。
AI 中文摘要
空间图描述了一个由相互作用的状态空间组成的系统。在这样的系统中，运动规划不仅仅是一个对象层面的问题，因为所选择的路径必须构成一个自然族。这引出了图的拓扑复杂度的概念，定义为它们的端点评估映射的截面范畴。在此设置中，点的作用由轨道扮演，即那些余极限为点的图。不同可容许轨道形状的选择导致了不同层次的范畴压缩。我们考虑了可表示轨道、离散轨道和任意拓扑轨道族，以及相应的轨道相对Lusternik–Schnirelmann范畴。它们与截面范畴的关系为图的拓扑复杂度提供了上界，而其值及适当的逆向极限提供了下界。对于由有限离散群索引的图，该构造恢复了等变LS-范畴、等变截面范畴和等变拓扑复杂度。
英文摘要
A diagram of spaces describes a system of interacting state spaces. Motion planning in such a system is not merely an objectwise problem since the selected paths must form a natural family. This leads to a notion of topological complexity for diagrams, defined as the sectional category of their endpoint evaluation map. The role of points in this setting is played by orbits, namely diagrams whose colimit is a point. Different choices of admissible orbit shapes give rise to different levels of categorical compression. We consider the families of representable, discrete and arbitrary topological orbits, and the corresponding orbit-relative Lusternik--Schnirelmann categories. Their relationship with sectional category provides upper bounds for the topological complexity of a diagram, while its values and suitable inverse limits provide lower bounds. For diagrams indexed by a finite discrete group, the construction recovers equivariant LS-category, equivariant sectional category and equivariant topological complexity.
Comments30 pages. Comments are welcome
