[论文解读] The geometry of oriented cubes

Iain R. Aitchison|arXiv (Cornell University)|Aug 10, 2010

Homotopy and Cohomology in Algebraic Topology参考文献 5被引用 10

一句话总结

本文提出了一种定向立方体的几何构造，为非交换上同调中的立方体上链条件提供了自然的归纳框架。它证明了杨-巴克斯方程、五边形方程以及高阶单形方程，均是同一高维立方体结构的不同表现形式。

ABSTRACT

This reports on the fundamental objects revealed by Ross Street, which he called `orientals'. Street's work was in part inspired by Robert's attempts to use N-category ideas to construct nets of C*-algebras in Minkowski space for applications to relativistic quantum field theory: Roberts' additional challenge was that `no amount of staring at the low dimensional cocycle conditions would reveal the pattern for higher dimensions'. This report takes up this challenge, presenting a natural inductive construction of explicit cubical cocyle conditions, and gives three ways in which the simplicial ones can be derived from these. (A dual string-diagram version of this work, giving rise to a Pascal's triangle of diagrams for cocycle conditions, has been described elsewhere by Street). A consequence of this work is that the Yang-Baxter equation, the `pentagon of pentagons', and higher simplex equations, are in essence different manifestations of the same underlying abstract structure. There has been recent interest in higher-categories, by computer scientists investigating concurrency theory, as well as by physicists, among others. The dual `string' version of this paper makes clear the relationship with higher-dimensional simplex equations in physics. Much work in this area has been done since these notes were written: no attempt has been made to update the original report. However, all diagrams have been redrawn by computer, replacing all original hand-drawn pictures.

研究动机与目标

• 为回应罗伯特提出的挑战：低维上链条件无法揭示高维中的模式。
• 通过n-立方体上的显式结构，为斯垂特的单纯定向体提供几何解释。
• 从立方体上链条件推导出单纯上链条件，建立立方体与单形之间的对偶性。
• 通过n-立方体边界中嵌套的嵌入圆盘序列，为非交换上同调定义规范的上链条件。
• 揭示杨-巴克斯方程、五边形之五边形与高阶单形方程背后深层的结构统一性。

提出的方法

• 将n维立方体构造为一个几何对象，其有序子立方体由{−, 0, +}^n中的元素标记。
• 通过将边界递归分解为半球面，定义n-立方体的k维源与目标。
• 通过在边界固定的半球面上对数据进行连续形变，来建模态射及高阶态射。
• 应用归纳构造，为所有维度生成显式的立方体上链条件。
• 通过立方体与单形结构之间的几何对应关系，从立方体条件推导出单纯条件。
• 引入子立方体的规范排序，以确保在各维度间源与目标分配的一致性。

实验结果

研究问题

• RQ1 如何系统性地揭示低维情况之外的高维上链条件的模式？
• RQ2 n-立方体上的何种几何结构编码了n-范畴的相干性数据？
• RQ3 源自n-单形的单纯定向体与n-立方体上的定向体结构有何关联？
• RQ4 杨-巴克斯方程与五边形方程以何种方式作为同一基础立方体结构的不同表现形式出现？
• RQ5 能否通过n-立方体边界中嵌套的嵌入圆盘序列，构建非交换上同调的几何解释？

主要发现

• 本文提供了立方体上链条件的自然归纳构造，其推广了单纯情形。
• 它确立了杨-巴克斯方程、五边形方程与高阶单形方程均为同一抽象结构在高阶范畴中的不同实现。
• 几何构造表明，通过其边界半球面的嵌套有序形变，n-立方体支持一个一致的n-范畴结构。
• 斯垂特的单纯定向体可通过对偶性从立方体定向体导出，且保持底层的相干性数据。
• n-立方体中子立方体的规范排序，为每个k ≤ n的维度提供了明确定义的源与目标系统。
• 本工作通过将数据扩展建模为边界固定的连续形变，为非交换上同调提供了几何基础。