On the additive image of 0th persistent homology
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914290343084032 |
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| author | Bauer, Ulrich Botnan, Magnus Bakke Oppermann, Steffen Steen, Johan |
| author_facet | Bauer, Ulrich Botnan, Magnus Bakke Oppermann, Steffen Steen, Johan |
| contents | For $X$ a finite category and $F$ a finite field, we study the additive image of the functor $\operatorname{H}_0(-,F) \colon \operatorname{rep}(X, \mathbf{Top}) \to \operatorname{rep}(X, \mathbf{Vect}_F)$, or equivalently, of the free functor $\operatorname{rep}(X, \mathbf{Set}) \to \operatorname{rep}(X, \mathbf{Vect}_F)$.
We characterize all finite categories $X$ for which the indecomposables in the additive image coincide with the indecomposable indicator representations and provide examples of quivers of wild representation type where the additive image contains only finitely many indecomposables. Motivated by questions in topological data analysis, we conduct a detailed analysis of the additive image for finite grids. In particular, we show that for grids of infinite representation type, there exist infinitely many indecomposables both within and outside the additive image.
We develop an algorithm for determining if a representation of a finite category is in the additive image. In addition, we investigate conditions for realizability and the effect of modifications of the source category and the underlying field.
The paper concludes with a discussion of the additive image of $\operatorname{H}_n(-,F)$ for an arbitrary field $F$, extending previous work for prime fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_09132 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the additive image of 0th persistent homology Bauer, Ulrich Botnan, Magnus Bakke Oppermann, Steffen Steen, Johan Representation Theory Algebraic Topology For $X$ a finite category and $F$ a finite field, we study the additive image of the functor $\operatorname{H}_0(-,F) \colon \operatorname{rep}(X, \mathbf{Top}) \to \operatorname{rep}(X, \mathbf{Vect}_F)$, or equivalently, of the free functor $\operatorname{rep}(X, \mathbf{Set}) \to \operatorname{rep}(X, \mathbf{Vect}_F)$. We characterize all finite categories $X$ for which the indecomposables in the additive image coincide with the indecomposable indicator representations and provide examples of quivers of wild representation type where the additive image contains only finitely many indecomposables. Motivated by questions in topological data analysis, we conduct a detailed analysis of the additive image for finite grids. In particular, we show that for grids of infinite representation type, there exist infinitely many indecomposables both within and outside the additive image. We develop an algorithm for determining if a representation of a finite category is in the additive image. In addition, we investigate conditions for realizability and the effect of modifications of the source category and the underlying field. The paper concludes with a discussion of the additive image of $\operatorname{H}_n(-,F)$ for an arbitrary field $F$, extending previous work for prime fields. |
| title | On the additive image of 0th persistent homology |
| topic | Representation Theory Algebraic Topology |
| url | https://arxiv.org/abs/2501.09132 |