Szczarba's twisted shuffle and equivariant path homology of directed graphs

TL;DR

This paper extends Szczarba's twisted shuffle to marked simplicial sets with group actions, establishing a chain isomorphism for path complexes and applying it to equivariant path homology of directed graphs.

Contribution

It generalizes Szczarba's twisted shuffle to the marked setting, proving a chain isomorphism for path complexes and defining equivariant path homology for directed graphs.

Findings

Proves the chain isomorphism in the marked setting.

Defines equivariant path homology via a twisted tensor product.

Applies the theory to the Borel construction for directed graphs.

Abstract

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.