# The Lifting Properties of A-Homotopy Theory

## TL;DR

This paper develops lifting properties within A-homotopy theory, a discrete analogue of classical homotopy, and uses these properties to compute the fundamental group of a 5-cycle graph.

## Contribution

It introduces lifting properties for A-homotopy theory and applies them to compute the fundamental group of a cycle graph, offering an alternative to existing methods.

## Key findings

- Established lifting properties for A-homotopy theory.
- Computed the fundamental group of the 5-cycle graph.
- Provided an alternative approach to classical fundamental group calculations.

## 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].