Zig-Zag Modules: Cosheaves and K-Theory

Abstract: Persistence modules have a natural home in the setting of stratified spaces and constructible cosheaves. In this article, we first give explicit constructible cosheaves for common data-motivated persistence modules, namely, for modules that arise from zig-zag filtrations (including monotone filtrations), and for augmented persistence modules (which encode the data of instantaneous events). We then identify an equivalence of categories between a particular notion of zig-zag modules and the combinatorial entrance path category on stratified $\mathbb{R}$. Finally, we compute the algebraic $K$-theory of generalized zig-zag modules and describe connections to both Euler curves and $K_0$ of the monoid of persistence diagrams as described by Bubenik and Elchesen.