Explore a technical colloquium talk that delves into the development of functoriality concepts for persistent homology in applied topology. Learn about the challenges and recent progress in creating a categorical framework for comparing persistence modules when analyzing real-world data samples. Discover how researchers are working to establish "induced map" techniques using practically available data, moving beyond traditional approaches that require known maps of combinatorial encodings. Examine the intersection of theoretical mathematics and practical applications as the speaker discusses collaborative research efforts to formalize reasoning processes in topological data analysis. Gain insights into how persistent homology, through filtered Vietoris-Rips complexes and multi-scale combinatorial representations, serves as a fundamental tool for understanding topological spaces embedded in metric spaces.