The homology digraph of a preordered space
type: Item
instance of: scholarly article
title: The homology digraph of a preordered space
zbMATH Open document ID: 1548.55007
author: Catarina Faustino
author: Thomas Kahl
published in: Journal of Homotopy and Related Structures
publication date: 2024-09-27T00:00:00Z
review text: 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.
reviewed by: Philippe Gaucher
Mathematics Subject Classification ID: 55N35
Mathematics Subject Classification ID: 55U25
Mathematics Subject Classification ID: 68Q99
zbMATH DE Number: 7919895
zbMATH Keywords: homology digraph
zbMATH Keywords: directed homology
zbMATH Keywords: directional graded vector space
zbMATH Keywords: bilinear relation
zbMATH Keywords: Künneth theorem
MaRDI profile type: Publication
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DOI: 10.1007/S40062-024-00352-7
description: scientific article; zbMATH DE number 7919895