Abstract: Inspired by the GLMY path homology theory of directed graphs and its generalisations to quivers and marked categories, we associate a path chain complex to a marked simplicial set and define its path homology. In this talk, I will introduce a Borel-type construction for marked simplicial sets equipped with simplicial group actions and twisting functions. This construction is given by a marked version of the twisted Cartesian product using the box product.

A classical theorem of Szczarba states that the twisted shuffle map induces a quasi-isomorphism between the chain complex of a twisted Cartesian product and an associated twisted tensor product. In the marked setting, we prove that this map restricts to an isomorphism of path chain complexes. As an application, digraphs with group actions admit a natural Borel construction as a special case of our framework. This leads to a notion of equivariant path homology, which can be computed via an explicit twisted tensor product. This is joint work with Shing-Tung Yau.