# Optimising the topological information of the $A_\infty$-persistence groups

## TL;DR

This paper enhances persistent homology by incorporating $A_
Infty$-coalgebra structures to better capture topological features across filtrations, providing deeper insights into the evolution of homological subspaces.

## Contribution

It introduces methods to select $A_
Infty$-coalgebras along filtrations, improving the fidelity of $A_
Infty$-persistence groups in topological data analysis.

## Key findings

- $A_
Infty$-persistence reveals richer topological information.
- Choosing appropriate $A_
Infty$-coalgebras enhances persistence group fidelity.
- The topological meaning of $V$ clarifies the role of $A_
Infty$-persistence.

## Abstract

Persistent homology typically studies the evolution of homology groups $H_p(X)$ (with coefficients in a field) along a filtration of topological spaces. $A_\infty$-persistence extends this theory by analysing the evolution of subspaces such as $V := \text{Ker}\, {\Delta_n}_{| H_p(X)} \subseteq H_p(X)$, where $\{\Delta_m\}_{m\geq1}$ denotes a structure of $A_\infty$-coalgebra on $H_*(X)$. In this paper we illustrate how $A_\infty$-persistence can be useful beyond persistent homology by discussing the topological meaning of $V$, which is the most basic form of $A_\infty$-persistence group. In addition, we explore how to choose $A_\infty$-coalgebras along a filtration to make the $A_\infty$-persistence groups carry more faithful information.