Simplicial-like Identities for The Paths and The Regular Paths on Discrete Sets

TL;DR

This paper establishes simplicial-like identities for paths and regular paths on discrete sets, enhancing the theoretical foundation for path-homology in digraphs through weighted face and co-face maps.

Contribution

It introduces new simplicial-like identities for paths on discrete sets using weighted face and co-face maps, bridging simplicial homotopy theory and digraph path-homology.

Findings

Proves simplicial-like identities for paths on discrete sets

Develops weighted face and co-face maps for path spaces

Enhances understanding of path-homology in digraphs

Abstract

Simplicial identities play an important and fundamental role in simplicial homotopy theory. On the other hand, the study of the paths and the regular paths on discrete sets is the foundation for the path-homology theory of digraphs. In this paper, by investigating some weighted face maps and weighted co-face maps on the space of the paths as well as the space of the regular paths, we prove some simplicial-like identities for the paths and the regular paths on discrete sets.