Metric Limits in Categories with a Flow

Abstract: In topological data science, categories with a flow have become ubiquitous, including as special cases examples like persistence modules and sheaves. With the flow comes an interleaving distance, which has proven useful for applications. We give simple, categorical conditions which guarantee metric completeness of a category with a flow, meaning that every Cauchy sequence has a limit. We also describe how to find a metric completion of a category with a flow by using its Yoneda embedding. The overarching goal of this work is to prepare the way for a theory of convergence of probability measures on these categories with a flow.