  Our objective in this article is to show a possibly interesting structure of
homotopic nature appearing in persistent (co)homology. Assuming that the
filtration of the (say) simplicial set embedded in a finite dimensional vector
space induces a multiplicative filtration (which would not be a so harsh
hypothesis in our setting) on the dg algebra given by the complex of simplicial
cochains, we may use a result by T. Kadeishvili to get a unique (up to
noncanonical equivalence) A_infinity-algebra structure on the complete
persistent cohomology of the filtered simplicial (or topological) set. We then
provide a construction of a (pseudo)metric on the set of all (generalized)
barcodes (that is, of all cohomological degrees) enriched with the
A_infinity-algebra structure stated before, refining the usual bottleneck
metric, and which is also independent of the particular A_infinity-algebra
structure chosen (among those equivalent to each other). We think that this
distance might deserve some attention for topological data analysis, for it in
particular can recognize different linking or foldings patterns, as in the
Borromean rings. As an aside, we give a simple proof of a result relating the
barcode structure between persistent homology and cohomology. This result was
observed in a recent article by V. de Silva, D. Morozov and M.
Vejdemo-Johansson under some restricted assumptions, which we do not suppose.
