Time-Varying Data as Sheaves: an Invitation to Narratives
Abstract
Modern science and engineering increasingly rely on time-varying data, yet the mathematical tools used to model temporal phenomena are often developed within separate disciplines, obscuring common principles and limiting the transfer of ideas across fields. This chapter presents the theory of narratives, an abstract framework for time-varying objects of any mathematical kind that supports both theoretical investigations and applications. To illustrate this perspective, the chapter develops three vignettes, each illustrating a different research direction. The first addresses a general concern: What information loss can occur when switching between different representations of temporal data? The second concerns structural and algorithmic approaches: How can we systematically decompose time-varying data into simple pieces and obtain invariants describing its structural complexity? The third is an application to control theory: How can we model multi-agent systems with switching communication topologies? More important than any individual vignette, the central message of this invitation is that a suitable abstract perspective can organize and guide research across remarkably diverse mathematical and scientific domains.
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BibTeX
@misc{leal2026timevarying,
title = {{Time-Varying Data as Sheaves: an Invitation to Narratives}},
author = {Wilmer Leal and Benjamin Merlin Bumpus and Jana K. Nickel and Johan García and James Fairbanks and Warren Dixon},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09056},
eprint = {2609.09056},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09056},
}
RIS
TY - UNPB
TI - Time-Varying Data as Sheaves: an Invitation to Narratives
AU - Wilmer Leal
AU - Benjamin Merlin Bumpus
AU - Jana K. Nickel
AU - Johan García
AU - James Fairbanks
AU - Warren Dixon
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09056
UR - https://arxiv.org/abs/2609.09056
AB - Modern science and engineering increasingly rely on time-varying data, yet the mathematical tools used to model temporal phenomena are often developed within separate disciplines, obscuring common principles and limiting the transfer of ideas across fields. This chapter presents the theory of narratives, an abstract framework for time-varying objects of any mathematical kind that supports both theoretical investigations and applications. To illustrate this perspective, the chapter develops three vignettes, each illustrating a different research direction. The first addresses a general concern: What information loss can occur when switching between different representations of temporal data? The second concerns structural and algorithmic approaches: How can we systematically decompose time-varying data into simple pieces and obtain invariants describing its structural complexity? The third is an application to control theory: How can we model multi-agent systems with switching communication topologies? More important than any individual vignette, the central message of this invitation is that a suitable abstract perspective can organize and guide research across remarkably diverse mathematical and scientific domains.
ER -