Interleaving distances are used widely in topological data analysis (TDA) as a tool
for comparing topological signatures of datasets. The theory of interleaving distances
has been extended through various category-theoretic constructions, enabling its
usage beyond standard constructions of TDA, while clarifying certain observed
stability phenomena by unifying them under a common framework. Inspired by
metrics used in the field of statistical shape analysis, which are based on minimizing
energy functions over group actions, we define three new types of increasingly general
interleaving distances. Our constructions use ideas from the theories of monoidal
actions and 2-categories. We show that these distances naturally extend
the category with a flow framework of de Silva, Munch and Stefanou and
the locally persistent category framework of Scoccola, and we provide a
general stability result. Along the way, we give examples of distances that
fit into our framework which connect to ideas from differential geometry,
geometric shape analysis, statistical TDA and multiparameter persistent
homology.
This article is currently available only to
readers at paying institutions. If enough institutions subscribe to
this Subscribe to Open journal for 2026, the
article will become Open Access in early 2026. Otherwise, this
article (and all 2026 articles) will be available only to paid
subscribers.
