a generalization of lifting properties for directed graphs

This exposition generalizes the fundamental lifting properties of a-homotopy theory for graphs to a broader framework suitable for directed graphs (digraphs). by leveraging the concept of a directed semicovering—a weakening of the classical covering map condition—we introduce the µ-fundamental monoid, denoted aµ 1 (d, v0).this algebraic invariant serves as an analogue to the generalized fundamental group introduced by de oliveira and da silva. we systematically reconstruct the path lifting, homotopy lifting, and lifting criterion theorems within this new µ-homotopy theory, demonstrating that while uniqueness of lifts is forfeited, the existence properties remain, there by providing a powerful new tool for the classification and analysis of directed graph structures.