Categorical Foundations of Persistent Homology: Bridging Classical Topology and Topological Data Analysis with Applications

Abstract

This paper introduces a novel categorical framework that unifies classical algebraic topology with modern topological data analysis through the lens of category theory. We develop the theory of persistence categories as a natural generalization of persistence modules, establishing functorial relationships between classical topological invariants and their persistent counterparts. Our approach reveals deep connections between sheaf cohomology, spectral sequences, and multi-parameter persistence, providing a rigorous mathematical foundation for understanding the stability and structure of topological features in data. We prove that persistent homology can be viewed as a particular instance of a more general categorical construction that encompasses both classical and computational topology. Furthermore, we establish new stability theorems for categorical persistence and demonstrate how classical results in algebraic topology can be lifted to the persistent setting through appropriate functorial constructions. We present practical applications in data science, computational biology, and machine learning, demonstrating the effectiveness of our theoretical framework through concrete implementations and computational experiments.