A higher homotopic extension of persistent (co)homology
The authors consider the \(A_\infty\)-algebra structure on the cohomology of a dg algebra: the Rees dg algebra. They show that the cohomology obtained is precisely the complete persistent cohomology of the filtered simplicial set. This allows them to: {\parindent=6mm \begin{itemize}\item[-] define a new barcodes distance on the collection of all generalized barcodes to the \(A_\infty\)-algebra constructed before. They show that their metric generalizes the bottleneck distance and that it is invariant under quasi-isomorphisms of \(A_\infty\)-algebras, \item[-] prove, under more general hypotheses, the well known open problem that there is an equivalence between the restriction of the barcode of \(n\)-th type for the persistent cohomology and that of the \(n\)-th type for the persistent homology. \end{itemize}}
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