Persistence modules are the central algebraic object in topological data analysis. This motivates the study of the geometry of the space of persistence modules. We isolate an elegant coherence condition that guarantees the interpolation and extension of sets of persistence modules. This "higher interpolation" is a consequence of the existence of certain universal constructions. As an application, it allows one to compare Vietoris-Rips and Cech complexes built within the space of persistence modules. This is joint work with Vin de Silva and Vidit Nanda.