The Lifting Properties of A-Homotopy Theory
Abstract
In classical homotopy theory, two spaces are homotopy equivalent if one space can be continuously deformed into the other. This theory, however, does not respect the discrete nature of graphs. For this reason, a discrete homotopy theory that recognizes the difference between the vertices and edges of a graph was invented, called A-homotopy theory [1-5]. In classical homotopy theory, covering spaces and lifting properties are often used to compute the fundamental group of the circle. In this paper, we develop the lifting properties for A-homotopy theory. Using a covering graph and these lifting properties, we compute the fundamental group of the 5-cycle, giving an alternate approach to [4].
Keywords
Cite
@article{arxiv.1904.12065,
title = {The Lifting Properties of A-Homotopy Theory},
author = {Rachel Hardeman Morrill},
journal= {arXiv preprint arXiv:1904.12065},
year = {2022}
}
Comments
32 pages, 14 figures, updated version. Revisions to all sections to incorporate helpful suggestions from referees which clarify several proofs and highlight original contributions. Minor changes throughout for grammar/style. In the previous update, the title changed from "Computing A-Homotopy Groups Using Coverings and Lifting Properties" to "The Lifting Properties of A-Homotopy Theory"