Data and homotopy types
arXiv:1908.06323
Abstract
This paper presents explicit assumptions for the existence of interleaving homotopy equivalences of both Vietoris-Rips and Lesnick complexes associated to an inclusion of data sets. Consequences of these assumptions are investigated on the space level, and for corresponding hierarchies of clusters and their sub-posets of branch points. Hierarchy posets and branch point posets admit a calculus of least upper bounds, which is used to show that the map of branch points associated to the inclusion of data sets is a controlled homotopy equivalence.
Preliminary version, 12 pages