What is a time-varying graph, or a time-varying topological space and more generally what does it mean for a mathematical structure to vary over time? Here we introduce categories of narratives: powerful tools for studying temporal graphs and other time-varying data structures. Narratives are sheaves on posets of intervals of time which specify snapshots of a temporal object as well as relationships between snapshots over the course of any given interval of time. This approach offers two significant advantages. First, when restricted to the base category of graphs, the theory is consistent with the well-established theory of temporal graphs, enabling the reproduction of results in this field. Second, the theory is general enough to extend results to a wide range of categories used in data analysis, such as groups, topological spaces, databases, Petri nets, simplicial complexes and many more. The approach overcomes the challenge of relating narratives of different types to each other and preserves the structure over time in a compositional sense. Furthermore our approach allows for the systematic relation of different kinds of narratives. In summary, this theory provides a consistent and general framework for analyzing dynamic systems, offering an essential tool for mathematicians and data scientists alike.

翻译：何为时变图，或时变拓扑空间，更一般地，数学结构随时间变化意味着什么？本文引入叙事范畴：研究时序图及其他时变数据结构的有力工具。叙事是时间区间偏序集上的层，它指定时变对象的快照以及任意给定时间区间内快照之间的关系。该方法具有两个显著优势：首先，当限制在图基范畴时，该理论与完善的时序图理论一致，能再现该领域的结果；其次，该理论具有足够的普适性，可将结果推广至数据分析中广泛使用的各类范畴，如群、拓扑空间、数据库、佩特里网、单纯复形等。该方法克服了不同类型叙事之间相互关联的难题，并以组合方式保留随时间演化的结构。此外，我们的方法还能系统关联不同种类的叙事。总之，该理论为分析动态系统提供了一致且通用的框架，成为数学家和数据科学家的重要工具。