Szczarba's twisted shuffle and equivariant path homology of 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.