Work

On the additive image of zeroth persistent homology

Year: 2026

Type: article

Abstract: For a 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.

Source: Transactions of the London Mathematical Society

DOI: 10.1112/tlm3.70027

Authors: Ulrich Bauer, Magnus B. Botnan, Steffen Oppermann, Johan Steen

Institutions: Munich Center for Machine Learning, Technical University of Munich, Vrije Universiteit Amsterdam, Norwegian University of Science and Technology

Language: English

Topic: Algebraic structures and combinatorial models

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $1,900

Funder: Deutsche Forschungsgemeinschaft