I am working in the area of topological data analysis (TDA). TDA has recently received a lot of attention both from the applied and the pure mathematics communities and it has shown a high potential of growth. My research program applies the machinery of higher category theory and homological algebra to the development of new methods for TDA and further unify existing methods. In particular, my work provides a generalized framework for the interleaving distance—a tool which is used as a dissimilarity measure for datasets—and a bridge for integrating more applied concepts such as dynamics, metrizability and stability with more pure areas such as algebra and sheaf theory. My Thesis studies the notion of dynamics on categories and develops a theory of interleaving distance on categories with a flow, thus providing a bridge between TDA and Dynamics.