Metrics for Generalized Persistence Modules

We consider the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets. Our constructions are functorial, which implies a form of stability for these metrics. We describe a large class of examples, inverse-image persistence modules, which occur whenever a topological space is mapped to a metric space. Several standard theories of persistence and their stability can be described in this framework. This includes the classical case of sublevelset persistent homology. We introduce a distinction between 'soft' and 'hard' stability theorems. While our treatment is direct and elementary, the approach can be explained abstractly in terms of monoidal functors.