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On the additive image of 0th persistent homology

Ulrich Bauer, Magnus Bakke Botnan, Steffen Oppermann, Johan Steen

TL;DR

This work classifies the additive image of H_0(-,F) for representations of finite categories X, showing it coincides with the additive image of the free functor on Set. It provides a complete framework to decide which representations lie in this image, and characterizes when indecomposables are indicator representations, when only finitely many indecomposables occur, and when the image is additively surjective. The authors develop algorithms for membership testing, analyze how realizability behaves under field extensions, subcategory operations, and edge contractions, and extend the analysis to quiver types (Dynkin and certain extended Dynkin) and finite grids. They further connect these results to higher homologies, proving that additively realizable representations for H_n(-;F) are characterized by algebraicity of matrix entries, and discuss implications for multiparameter persistence and topological data analysis. The results illuminate how orientation and field choice crucially affect the additive image and provide practical tools for identifying realizable representations in TDA contexts.

Abstract

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.

On the additive image of 0th persistent homology

TL;DR

This work classifies the additive image of H_0(-,F) for representations of finite categories X, showing it coincides with the additive image of the free functor on Set. It provides a complete framework to decide which representations lie in this image, and characterizes when indecomposables are indicator representations, when only finitely many indecomposables occur, and when the image is additively surjective. The authors develop algorithms for membership testing, analyze how realizability behaves under field extensions, subcategory operations, and edge contractions, and extend the analysis to quiver types (Dynkin and certain extended Dynkin) and finite grids. They further connect these results to higher homologies, proving that additively realizable representations for H_n(-;F) are characterized by algebraicity of matrix entries, and discuss implications for multiparameter persistence and topological data analysis. The results illuminate how orientation and field choice crucially affect the additive image and provide practical tools for identifying realizable representations in TDA contexts.

Abstract

For a finite category and a finite field, we study the additive image of the functor , or equivalently, of the free functor . We characterize all finite categories 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 for an arbitrary field , extending previous work for prime fields.
Paper Structure (33 sections, 62 theorems, 85 equations, 3 figures)

This paper contains 33 sections, 62 theorems, 85 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Homology in degree 0
  3. Main question
  4. Overview and Structure
  5. Related Work
  6. First properties
  7. Setting the stage
  8. The additive image over pointed sets
  9. Field extensions
  10. Determining membership in the additive image
  11. Running time for finite categories
  12. Scalars of finite order
  13. Removing scalars via Vandermonde matrices
  14. Sets with group actions
  15. An improved algorithm for testing membership in the additive image
  16. ...and 18 more sections

Key Result

Theorem 1.1

Linear representations of $A_{m_1}\otimes A_{m_2}$ that are epimorphic in the horizontal direction correspond (up to a finitely classified, completely explicit list of direct summands) to representations of $A_{m_1}\otimes A_{m_2-1}$.

Figures (3)

  • Figure 1: A bifiltration of a topological space.
  • Figure 2: Pointed set realizations for the preprojective representations of $\tilde{D}_4$ with two ingoing and two outgoing arrows; see Proposition \ref{['prop.preproj_and_preinj_are_induced']}. The linearized representation obtained from the top-most diagram is shown to be indecomposable in Example \ref{['ex.D4.preproj.indec']}.
  • Figure 3: Pointed set realizations for the preinjective representations of $\tilde{D}_4$ with two ingoing and two outgoing arrows; see Proposition \ref{['prop.preproj_and_preinj_are_induced']}.

Theorems & Definitions (132)

  • Theorem 1.1: bauer2020cotorsion
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Lemma 3.5
  • ...and 122 more