A categorical study of persistent homology for closure space

Abstract

We begin with the concept of closure space and show how it relates to other objects like graphs and metric spaces in categorical terms. We also study the notion of simplicial sets and show that there is an adjunction between the category of simplicial sets and the category of closure spaces. We will define the homology and cohomology of simplicial sets and apply that treatment to the construction of various homologies and cohomologies for closure space. Moreover, we also present an investigation of the Dold-Kan correspondence theorem. Finally, we will focus on the categorical foundation of persistent homology and promote a general formalization of a stability theorem; both of these are at the heart of topological data analysis.