Zig-Zag Modules: Cosheaves and K-Theory
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866917847995777024 |
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| author | Grady, Ryan E. Schenfisch, Anna |
| author_facet | Grady, Ryan E. Schenfisch, Anna |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_04591 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Zig-Zag Modules: Cosheaves and K-Theory Grady, Ryan E. Schenfisch, Anna Algebraic Topology Computational Geometry Category Theory 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. |
| title | Zig-Zag Modules: Cosheaves and K-Theory |
| topic | Algebraic Topology Computational Geometry Category Theory |
| url | https://arxiv.org/abs/2110.04591 |