On the additive image of 0th persistent homology

TL;DR

This paper characterizes the additive image of 0th persistent homology functors over finite categories, providing structural insights, algorithms, and examples relevant to topological data analysis.

Contribution

It offers a complete characterization of indecomposables in the additive image for finite categories and develops an algorithm to identify representations in this image.

Findings

Indecomposables in the additive image coincide with indicator representations for certain categories.

Finite grids of infinite representation type have infinitely many indecomposables in the additive image.

An algorithm is provided to determine if a representation lies in the additive image.

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.