Persistent homology typically studies the evolution of homology groups $$H_p(X)$$ H p ( X ) (with coefficients in a field) along a filtration of topological spaces. $$A_\infty $$ A ∞ -persistence extends this theory by analysing the evolution of subspaces such as $$V :=\text {Ker}\,{\Delta _n}_{| H_p(X)} \subseteq H_p(X)$$ V : = Ker Δ n | H p ( X ) ⊆ H p ( X ) , where $$\{\Delta _m\}_{m\ge 1}$$ { Δ m } m ≥ 1 denotes a structure of $$A_\infty $$ A ∞ -coalgebra on $$H_*(X)$$ H ∗ ( X ) . In this paper we illustrate how $$A_\infty $$ A ∞ -persistence can be useful beyond persistent homology by discussing the topological meaning of V, which is the most basic form of $$A_\infty $$ A ∞ -persistence group. In addition, we explore how to choose $$A_\infty $$ A ∞ -coalgebras along a filtration to make the $$A_\infty $$ A ∞ -persistence groups carry more faithful information.