Szczarba的扭曲洗牌与有向图的等变路径同调
Szczarba's twisted shuffle and equivariant path homology of directed graphs
AI总结：
本文研究了标记设定下路径同调的等变性质，通过扭曲洗牌定义了链同构，并应用于有向图的Borel构造。
摘要：
针对一个标记的简单集，可以关联其路径链复形，并定义其同调为其复形的同调，受有向图、有向图和标记范畴的路径同调理论启发。给定一个带有简单群作用的标记简单集，该作用保持标记和退化1维简单集，并伴有扭曲函数，我们定义了一个标记的扭曲笛卡尔积。经典上，Szczarba的扭曲洗牌提供了扭曲笛卡尔积链复形与相应扭曲张量积之间的一个准同构。在本论文中，我们证明在标记设定下，该映射限制为路径链复形上的链同构。作为应用，对于带有群作用的有向图，我们获得自然的Borel构造作为标记扭曲笛卡尔积的特例。等变路径同调被定义为该构造的同调，并通过显式的扭曲张量积进行计算。
英文摘要：
To a marked simplicial set one can associate its path chain complex, and define its homology to be the homology of this complex, inspired by path homology theories for directed graphs, quivers, and marked categories. Given a marked simplicial set with a simplicial group action preserving the markings and degenerate 1-simplices, together with a twisting function, we define a marked twisted Cartesian product using the box product. Classically, Szczarba's twisted shuffle provides a quasi-isomorphism between the chain complex of a twisted Cartesian product and the corresponding twisted tensor product. In this paper, we prove that in the marked setting, this map restricts to a chain isomorphism on path chain complexes. As an application, for directed graphs with group actions, we obtain a natural Borel construction as a special case of marked twisted Cartesian products. Equivariant path homology is defined as the homology of this construction and is computed by an explicit twisted tensor product.
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