Lifting Properties for A-Homotopy Theory

In classical homotopy theory, two spaces are considered 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 \cite{Atkin1, Atkin2, BabsonHomotopy, BarceloFoundations, BarceloPerspectives}. 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 cycle $C_{5}$, giving an alternate approach to [4].