Towards a Unified Theory of Time-Varying Data

TL;DR

The paper addresses the lack of a unifying theory for time-varying data by introducing narratives—sheaf- and cosheaf-based constructions on posets of time intervals—as an object-agnostic framework that can model any category with limits and colimits. It formalizes persistent and cumulative perspectives as dual categorical structures, connected by an adjunction, and provides a systematic method to lift static notions to temporal analogues via change-of-base and temporal-resolution functors. The main contributions include the precise definitions of Pe and Cu, the adjunction $\\mathscr{K} \\dashv \\mathscr{P}$, and the demonstration that standard temporal graphs and other structures (e.g., Petri nets, metric spaces) fit naturally within this unified framework, enabling morphisms and dynamic-type analyses. By bridging data-centric and dynamics-centric viewpoints, the work lays a foundation for a general theory of temporal data with potential impact on temporal graph theory, TDA, and dynamical systems modeling, while highlighting avenues for future exploration and cross-domain applications.

Abstract

What is a time-varying graph, a time-varying topological space, or, more generally, a mathematical structure that evolves over time? In this work, we lay the foundations for a general theory of temporal data by introducing categories of narratives. These are sheaves on posets of time intervals that encode snapshots of a temporal object along with the relationships between them. This theory satisfies five desiderata distilled from the burgeoning field of time-varying graphs: (D1) it defines both time-varying objects and their morphisms; (D2) it distinguishes between cumulative and persistent interpretations and provides principled methods for transitioning between them; (D3) it systematically lifts static notions to their temporal analogues; (D4) it is object agnostic; (D5) it integrates with theories of dynamical systems. To achieve this, we build upon existing categorical and sheaf-theoretic approaches to temporal graph theory, generalizing them to any category with limits and colimits. We also formalize tacit intuitions that, while present, often remain implicit in temporal graph theory. Beyond synthesizing and reformulating existing ideas in categorical language, we introduce sheaf-theoretic constructions and prove results that, to our knowledge, have not appeared in the temporal data literature - such as the adjunction between persistent and cumulative narratives. More importantly, we integrate these existing and novel elements into a consistent and coherent framework, setting the stage for a unified theory of time-varying data.

Figures (2)

• Figure 1: A schematic visualization of a sheaf on a discrete time category (a persistent narrative) with three snapshots. The domain is the time category $\mathsf{I}_{\mathbb{N}}/[1,3]$, a join-semi-lattice whose objects are closed subintervals of $[1,3]$ and whose morphisms are interval inclusions. The codomain of the sheaf is a category $\mathsf{D}$ with pullbacks. We use the shorthand $F_i^j$ (for $i \leq j$) to denote the data assigned the interval $[i,j]$ by $F$ (i.e. $F_i^j := F([i,j])$).
• Figure 2: A temporal graph along with its persistent and cumulative narratives

Theorems & Definitions (26)

• Definition 2.1
• Definition 2.2: Interval categories schultz2020dynamical
• Definition 2.3: Closed Intervals and Inclusions
• Lemma 2.5
• Definition 2.6
• Proposition 2.7: $\mathsf{T}$-sheaves and $\mathsf{T}$-cosheaves
• proof
• Definition 2.8
• Proposition 2.9
• proof