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 R n induces a multiplicative filtration on the dg algebra of simplicial cochains, we use a result by Kadeishvili to get a unique A ∞ -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 ∞ -algebra structure stated before, refining the usual bottleneck metric, and which is also independent of the particular A ∞ -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.

EB00000003630K

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