A higher homotopic extension of persistent (co)homology

Our objective is to show a possibly interesting structure of homotopic nature appearing in persistent (co)homology. Assuming that the filtration of a simplicial set embedded in \({\mathbb {R}}^{n}\) induces a multiplicative filtration on the dg algebra of simplicial cochains, we use a result by Kadeishvili to get a unique \(A_{\infty }\)-algebra structure on the complete persistent cohomology of the filtered simplicial set. We then construct of a (pseudo)metric on the set of all barcodes of all cohomological degrees enriched with the \(A_{\infty }\)-algebra structure stated before, refining the usual bottleneck metric, and which is also independent of the particular \(A_{\infty }\)-algebra structure chosen. We also compute this distance for some basic examples. As an aside, we give a simple proof of a result relating the barcode structure between persistent homology and cohomology, that was observed by de Silva, Morozov, and Vejdemo-Johansson under some restricted assumptions which we do not suppose.