The homology digraph of a preordered space (Q6612008)

In this paper, the authors introduce a novel concept in the field of directed algebraic topology, called the homology digraph. This concept is specifically designed for preordered spaces, which are spaces equipped with a preorder relation that represents a directional structure. The authors demonstrate that the homology digraph is a directed homotopy invariant, meaning it remains unchanged under directed homotopy equivalences. Note that by directed homotopy equivalences, the authors mean a monotone map of preordered spaces that is a homotopy equivalence with a monotone homotopy inverse. One of the contributions of this work is the establishment of variants of the main results of ordinary singular homology theory for the homology digraph. In particular, the authors prove a Künneth formula for the homology digraph. This formula allows for the computation of the homology digraph of a product of preordered spaces from the homology digraphs of the individual components. This result provides a method to break down complex computations into simpler, more manageable parts. The paper situates itself within the broader context of directed algebraic topology, which has applications in concurrency theory -- a domain of theoretical computer science that deals with systems of simultaneously executing processes. The state space of a concurrent system can be modeled as a directed space, and the executions of the system are represented by directed paths. The homology digraph provides a new tool for analyzing such systems.