  The interleaving distance was originally defined in the field of Topological
Data Analysis (TDA) by Chazal et al. as a metric on the class of persistence
modules parametrized over the real line. Bubenik et al. subsequently extended
the definition to categories of functors on a poset, the objects in these
categories being regarded as `generalized persistence modules'. These metrics
typically depend on the choice of a lax semigroup of endomorphisms of the
poset. The purpose of the present paper is to develop a more general framework
for the notion of interleaving distance using the theory of `actegories'.
Specifically, we extend the notion of interleaving distance to arbitrary
categories equipped with a flow, i.e. a lax monoidal action by the monoid
$[0,\infty)$. In this way, the class of objects in such a category acquires the
structure of a Lawvere metric space. Functors that are colax
$[0,\infty)$-equivariant yield maps that are $1$-Lipschitz. This leads to
concise proofs of various known stability results from TDA, by considering
appropriate colax $[0,\infty)$-equivariant functors. Along the way, we show
that several common metrics, including the Hausdorff distance and the
$L^{\infty}$-norm, can be realized as interleaving distances in this general
perspective.
