#### Abstract

The persistence diagram is very different in philosophy from the barcode. Suppose we have a constructible persistence module of vector spaces. Its barcode is its list of indecomposables. Its persistence diagram is an encoding of all persistent vector spaces. In the setting of vector spaces, we know that these two notions are equivalent. However, we quickly run into problems if we try to generalize the barcode beyond the setting of vector spaces. In this talk, I will generalize the persistence diagram to the setting of constructible persistence modules valued in any symmetric monoidal category. For example, the category of sets, the category of vector spaces, and the category of abelian groups are symmetric monoidal categories. As an immediate consequence, we can finally talk about persistent homology over integer coefficients!