Rectifications of homotopy interleaving distances

Topological data analysis employs an `interleaving distance’ between persistence spaces as a means of measuring when two such objects are approximately isomorphic. By generalizing this notion to homotopy interleavings, one can instead measure when persistent spaces are approximately weakly equivalent. Michael Lesnick and Andrew Blumberg developed a universal homotopy interleaving distance and conjectured that it differed from the interleaving distance on a homotopy category by at most a multiplicative constant. The conjecture was later proved by Edoardo Lanari and Luis Scoccola. I gave motivation and proof of this conjecture, stating the homotopy interleaving distance on \(\textnormal{Fun}(\mathbb{Z},Spc)\) is within a multiplicative constant of the interleaving distance on \(\textnormal{Fun}(\mathbb{Z},\textnormal{Ho}(Spc))\). I also detailed the failure of this phenomenon for multi-persistent spaces.