The Representation Theorem of Persistent Homology Revisited and Generalized

The Representation Theorem by Zomorodian and Carlsson has been the starting point of the study of persistent homology under the lens of algebraic representation theory. In this work, we give a more accurate statement of the original theorem and provide a complete and self-contained proof. Furthermore, we generalize the statement from the case of linear sequences of $R$-modules to $R$-modules indexed over more general monoids. This generalization subsumes the Representation Theorem of multidimensional persistence as a special case.