ADVANCES IN ALGEBRAIC TOPOLOGY: PERSISTENT HOMOLOGY, HIGHER CATEGORY THEORY, AND APPLICATIONS TO MODERN GEOMETRIC STRUCTURES

Abstract

This study presents a theoretical synthesis that connects algebraic topology, persistent homology, and higher category theory to provide a unified mathematical perspective for understanding geometric and algebraic structures. The motivation lies in advancing the conceptual foundations of topology by integrating the multi-scale analysis of persistent homology with the hierarchical relational framework of higher categories. The paper proposes a higher categorical interpretation of persistence modules, in which morphisms and higher morphisms encode the evolution and interaction of topological features across filtration levels. A bifunctorial framework is developed to connect filtered complexes with higher morphism spaces, enabling the representation of topological persistence within categorical vector spaces. This structural formulation enriches the study of modern geometric frameworks such as manifolds, orbifolds, and derived geometries, and provides a categorical foundation for modelling transformations in topological quantum field theories. The research emphasizes interpretability, consistency, and abstraction rather than computational experimentation. By embedding persistent homology within higher categorical contexts, algebraic topology gains an extended capacity to model, compare, and classify complex geometric and algebraic relationships, contributing to the broader development of theoretical mathematics.