Acting Fibrations and Lifting Functions in the Homotopy Theory of Single Intersection Graphs over Topological Semigroups

In this paper, we study the homotopy theory of single intersection graphs arising from acting spaces over topological semigroups. An acting space (S,B) is defined as a topological space B equipped with a continuous action of a topological semigroup S, generalizing the notion of algebraic actions in a topological setting. To connect this structure with graph theory, we associate to each acting space a single intersection graph GSB, whose vertices are proper SB-subacting spaces, and two vertices are adjacent if their intersection is a singleton set. This graph construction encodes both algebraic and topological interactions between subacting spaces and provides a framework to study connectivity and homotopical properties via combinatorial methods. We then work within a categorical framework, where objects are graphical acting semigroups and morphisms are S-acting maps, allowing us to systematically study structural properties and their invariance under morphisms. In this setting, we introduce the notion of acting fibrations and formulate the corresponding lifting problem. Our main result establishes that an S-acting map is an acting fibration if and only if it admits an A-lifting function, providing a characterization analogous to classical fibration theory. Furthermore, we introduce A-regular lifting functions and analyze their role in preserving homotopical structures, including a natural homotopy extension property.