A metrically complete and Krull--Schmidt space of multiparameter persistence modules

We show that the observable category of q-tame multiparameter persistence modules satisfies good metric and algebraic properties: it forms a complete metric space with respect to the interleaving distance, and it is Krull--Schmidt in the sense that every object admits an essentially unique decomposition into indecomposables. Moreover, we show that these metric and algebraic structures are compatible: two objects are at distance zero if and only if they are isomorphic. We argue that the observable category of q-tame multiparameter persistence modules is the right setup for multiparameter persistence by showing that many of the categories already considered in the literature form full subcategory of this category. We also characterize precompact sets in terms of finite representation type of certain discretizations, and show that the image of several of the main constructions in multiparameter persistence is precompact.