  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.
