Towards a Unified Theory of Time-Varying Data

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 sow the seeds of a general theory of temporal data by introducing categories of narratives. These are sheaves on posets of intervals of time which specify snap-shots of a temporal object as well as relationships between them. This theory satisfies five desiderata distilled from the burgeoning field of time-varying graphs: (D1) any theory of temporal data should define not only time-varying objects, but also appropriate morphisms thereof; (D2) in contrast to being a mere sequence, temporal data should explicitly record whether it is to be viewed cumulatively or persistently. Furthermore there should be methods of conversion between these two viewpoints; (D3) any theory of temporal data should come equipped with systematic ways of lifting static notions to their appropriate temporal analogues; (D4) theories of temporal data should be object agnostic and applicable to any mathematical structure; (D5) any theory of temporal data should be seamlessly interoperable with theories of dynamical systems. In summary, our theory of narratives provides a consistent and general framework for studying mathematical structures which change over time. This is a first step towards a unified theory of time-varying data.

Join work with: Benjamin Merlin Bumpus, James Fairbanks, Martti Karvonen, and Frédéric Simard