On the Structural Theorem of Persistent Homology

Abstract

We study the categorical framework for the computation of persistent homology, without reliance on a particular computational algorithm. The computation of persistent homology is commonly summarized as a matrix theorem, which we call the Matrix Structural Theorem. Any of the various algorithms for computing persistent homology constitutes a constructive proof of the Matrix Structural Theorem. We showthat the Matrix Structural Theorem is equivalent to the Krull-Schmidt property of the category of filtered chain complexes. We separately establish the Krull-Schmidt property by abstract categorical methods, yielding a novel nonconstructive proof of the Matrix Structural Theorem. These results provide the foundation for an alternate categorical framework for decomposition in persitent homology, bypassing the usual persistence vector spaces and quiver representations.