Amplitudes on abelian categories

Abstract

The use of persistent homology in applications is justified by the validity of certain stability results. At the core of such results is a notion of distance between the invariants that one associates to data sets. While such distances are well-understood in the one-parameter case, the situation for multiparameter persistence modules is more challenging, since there exists no generalisation of the barcode. Here we introduce a general framework to study stability questions in multiparameter persistence. We introduce amplitudes -- invariants that arise from assigning a non-negative real number to each persistence module, and which are monotone and subadditive in an appropriate sense -- and then study different ways to associate distances to such invariants. Our framework is very comprehensive, as many different invariants that have been introduced in the Topological Data Analysis literature are examples of amplitudes, and furthermore many known distances for multiparameter persistence can be shown to be distances from amplitudes. Finally, we show how our framework can be used to prove new stability results.Comment: 49 pages, major revisions throughout; added section on preliminaries, reorganised/improved main sections in the paper, removed discussion about general finitely encoded module