  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.
