Exact weights, path metrics, and algebraic Wasserstein distances (Q6175710)

A \textit{weight} in an abelian category \(\mathbf{A}\) assigns to each object \(A\) a number \(w(A) \in [0, \infty]\) vanishing on the 0 object and invariant under isomorphisms. A \textit{zigzag} from \(A\) to \(B\) in \(\mathbf{A}\) is a sequence of morphisms \(A = A_0 \rightarrow A_1 \leftarrow A_2 \rightarrow \ldots \leftarrow A_n = B\). This paper defines the \textit{cost} of a zigzag as the sum of weights of kernels and cokernels along it; it then defines the \textit{path metric} by minimizing cost among all zigzags from \(A\) to \(B\). Conversely, a given metric on \(\mathbf{A}\) provides a weight. A notion of \textit{stability} of a weight and a condition by which a weight \textit{bounds its path metric} are defined. Further, a weight is said to be \textit{exact} if, for each short exact sequence, the weight of each term (extremes excluded) does not exceed the sum of the weights of the other two (dimension is an example of such a weight in the category of \(K\)-vector spaces). The main general theorem of the paper (Thm. 1.1, better specified as Thm. 3.28) states that a weight is exact if and only if it is stable if and only if it bounds its path metric. The paper then dives into generalized persistence. Classically [\textit{A. Zomorodian} and \textit{G. Carlsson}, Discrete Comput. Geom. 33, No. 2, 249--274 (2005; Zbl 1069.55003)], the main tool of persistent homology is the \textit{persistence module}, i.e. a functor from \((\mathbb{R}, \le)\) to the category of \(R\)-modules (\(R\) a fixed ring, mostly a field). There have been several generalizations, that change either domain or range or the very nature of the functor; to mention a few: [\textit{P. Bubenik} and \textit{J. A. Scott}, Discrete Comput. Geom. 51, No. 3, 600--627 (2014; Zbl 1295.55005); \textit{A. Patel}, J. Appl. Comput. Topol. 1, No. 3--4, 397--419 (2018; Zbl 1398.18015); \textit{M. G. Bergomi} and \textit{P. Vertechi}, Theory Appl. Categ. 35, 228--260 (2020; Zbl 1439.55006); \textit{W. Kim} and \textit{F. Mémoli}, J. Appl. Comput. Topol. 5, No. 4, 533--581 (2021; Zbl 1500.55004)]. Here, as persistence modules, the authors mean functors from a small category \(\mathbf{P}\) to an Abelian one \(\mathbf{A}\); with natural transformations as morphisms, they form an Abelian category. Given an exact weight \(w\) in \(\mathbf{A}\) and a measure \(\mu\) on the object set of \(\mathbf{P}\), an exact weight \(W\) is defined on persistence modules. Then, \textit{p-Wasserstein distances} are defined for this setting; their relations with the classical Wasserstein distances and with the path metric are established. The matrix representation of morphisms between persistence modules is studied in depth. Examples in multiparameter and zigzag persistence conclude the paper.