Homology, Homotopy and Applications

Volume 25 (2023)

Number 2

Zig-zag modules: cosheaves and $k$-theory

Pages: 243 – 274

DOI: https://dx.doi.org/10.4310/HHA.2023.v25.n2.a11


Ryan Grady (Department of Mathematical Sciences, Montana State University, Bozeman, Mt.)

Anna Schenfisch (Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands)


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.


persistence module, zig-zag persistence, cosheaf, algebraic $K$-theory

2010 Mathematics Subject Classification

Primary 18F25. Secondary 19M05, 32S60.

Anna Schenfisch is supported by the National Science Foundation under NIH/NSF DMS 1664858.

Received 4 April 2022

Received revised 23 July 2022

Accepted 21 September 2022

Published 1 November 2023