  The use of topological persistence in contemporary data analysis has provided
considerable impetus for investigations into the geometric and
functional-analytic structure of the space of persistence modules. In this
paper, we isolate a coherence criterion which guarantees the extensibility of
non-expansive maps into this space across embeddings of the domain to larger
ambient metric spaces. Our coherence criterion is category-theoretic, allowing
Kan extensions to provide the desired extensions. Our main construction gives
an isometric embedding of a metric space into the metric space of persistence
modules with values in the spacetime of this metric space. As a consequence of
such "higher interpolation", it becomes possible to compare Vietoris-Rips and
\v{C}ech complexes built within the space of persistence modules.
