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Time-Varying Data as Sheaves: an Invitation to Narratives
Wilmer Leal, Benjamin Merlin Bumpus, Jana K. Nickel, Johan García, James Fairbanks, Warren Dixon
TL;DR
The chapter addresses how fragmented mathematical approaches can model time-varying data across disciplines. It presents narratives as an abstract, object-agnostic framework and develops three vignettes on viewpoint changes, structured decompositions, and switching multi-agent systems. The framework organizes these directions while supporting extensions to temporal cellular-sheaf dynamics and stability questions.
Problem
Mathematical tools for time-varying phenomena remain fragmented across disciplines, motivating a unified framework for modeling temporal data and transferring ideas across domains.
Method
The chapter develops narratives as an abstract framework for time-varying objects, using categorical constructions and applications to persistence, decompositions, graph structure, and cellular sheaves.
Results
The three vignettes demonstrate narratives for analyzing information loss between persistent and cumulative viewpoints, temporalizing structured decompositions, and modeling switching communication systems.
Takeaways & Limitations
Narratives provide an object-agnostic perspective that organizes research on time-varying data across diverse mathematical and scientific applications.
Takeaways & Limitations
The chapter focuses on interval categories, while branching temporal models are identified as a direction for future research.
Abstract
from arXiv · showhide
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.
1 Introduction
The chapter introduces narratives as a unified, object-agnostic language for time-varying data, addressing fragmented temporal theories through three research vignettes. These vignettes study information loss between temporal viewpoints, structured decompositions, and switching multi-agent systems.
- The framework is designed to define temporal data and morphisms, distinguish cumulative and persistent perspectives, temporalize static concepts, remain object agnostic, and connect with dynamical systems.
- Together, the vignettes illustrate how an abstract perspective can organize research across diverse mathematical and scientific domains.
- Narratives provide a unified language for describing diverse questions about time-varying data.
- The first vignette analyzes information loss when converting between persistent and cumulative narratives through an adjunction that is not generally an equivalence.
- The second vignette lifts structured decompositions to persistent narratives so time-varying data can be decomposed while preserving temporal structure.
- The third vignette models changing multi-agent communication systems with narratives valued in cellular sheaves, recording topology, local state spaces, sensing maps, and compatibility constraints.
2 Background on Narratives: How to Model Time-Varying Data using Sheaves
Narratives model time-varying objects as sheaves or cosheaves over interval-based time categories, preserving relations across overlapping intervals. Their categorical structure supports changing temporal viewpoints, functorial data conversion, and temporalized graph constructions, while remaining extensible to other models of time.
- Narratives are sheaves or cosheaves over time categories, recording how temporal information relates across overlapping intervals.
- A time category is a sub-join-semilattice of closed intervals in continuous or discrete time, ordered by inclusion.
- The framework focuses on interval categories but can accommodate alternatives such as finite branching trees for branching temporal evolutions.
- Persistent narratives use sheaves to represent information that remains valid across intervals, whereas cumulative narratives use cosheaves to represent information aggregated over time.
- The functors K and P form an adjunction between persistent and cumulative narratives, but their round trips generally do not recover the original temporal structure.
- Narratives support principled temporalization by transporting constructions across data categories through functorial operations that preserve the relevant universal constructions.
- Graph case studies recover temporal paths and cliques as narrative subobjects or morphism characterizations, with categorical dualities depending on cumulative or persistent perspective.
3.1 Vignette 1: Narratives and the Fixed Points of the Persistence–Accumulation Adjunction
The persistence–accumulation adjunction can lose information when temporal data are converted between viewpoints, motivating narratives that encode both descriptions and their comparison. The chapter factorizes the adjunction through cotwisted-arrow-valued narratives and characterizes conditions under which round trips recover the original data.
- Information loss: The pushout–pullback round trip can fail to recover a persistent narrative because the unit ηF: F → P K F need not be an isomorphism.The persistent data are prescribed, whereas the round trip constructs a canonical approximation through pushouts and pullbacks.
- Fixed points: A fixed point is a persistent narrative whose unit ηF: F → P K F is an isomorphism, equivalently whose prescribed span equals the pullback of its pushout.This characterizes exact recovery in the persistent direction.
- Rigidity: In adhesive categories, spans of monomorphisms are fixed under pushout–pullback, but this does not imply the dual cospan property or right rigidity.The result applies broadly to double-pushout graph rewriting, while the converse remains unsupported.
- Information loss: The dual round trips can behave asymmetrically: cumulative data may be recovered even when persistent data are not.Figure 1 exhibits recovery after pullback–pushout but not after pushout–pullback.
- Narrative construction: Cotwisted-arrow-valued narratives simultaneously encode persistent and cumulative components together with their comparison, and their sheaf structure supports both pullback and pushout behavior.Objects are morphisms in D, while morphisms encode the corresponding factorization; when D has pullbacks and pushouts, the cotwisted arrow category has pullbacks.
- Narrative construction: The persistence–accumulation adjunction factorizes through the category of narratives, with componentwise domain and codomain functors recovering persistent and cumulative narratives.Canonical completion functors embed each viewpoint into narratives, and comparison morphisms measure agreement with the adjunction’s canonical reconstructions.
3.2 Vignette 2: Decomposing Time-Varying Data into Simple Pieces: Structured Decompositions of Narratives
Structured decompositions provide a systematic way to split time-varying data into evolving pieces while recording their overlaps, extending decomposition theories from static objects to persistent narratives.
- Motivation: Decompositions divide complex objects into smaller pieces and record their overlaps so the original object can be reconstructed by gluing.For temporal data, the pieces and their structure maps must also represent persistence, merging, splitting, or disappearance.
- Motivation: The proposed invariant measures a time-varying object’s structural complexity through the complexity of its pieces and their overlaps.The method avoids decomposing each snapshot independently, requiring the decomposition to capture evolution of the global structure.
- Temporalization: Narratives temporalize structured decompositions by lifting decomposition theories from static objects to persistent narratives.The framework uses spined structured decomposition categories, a category-theoretical generalization of tree-decompositions.
- Background: Structured decompositions generalize tree-decompositions and support established applications across graph theory, logic, algorithms, and complexity theory.Related uses include graph structure results and efficient dynamic programming on bounded-tree-width graphs.
- Tree-decompositions: A tree-decomposition consists of a tree indexed by vertex bags satisfying coverage and connectedness conditions for each graph vertex.The decomposition tree is the model, while the bags and induced subgraphs are its parts.
Tree-width.
Tree-width measures how closely a graph resembles a tree through the smallest possible width of its tree-decompositions, motivating a temporal generalization for evolving structures.
- Tree-width: Tree-width measures the extent to which a graph’s structure resembles a tree by minimizing the width of its tree-decompositions.Smaller decomposition parts correspond to more tree-like structure.
- Tree-width: The width of a tree-decomposition is the maximum bag size minus one, and tree-width is the minimum such width over all decompositions.The subtraction by one makes every tree have tree-width one.
- Structured decompositions: Spined structured decomposition categories provide a unified axiomatic framework that recovers tree-width and several generalized graph-width notions.Examples include complemented tree-width, tree independence number, hypergraph tree-width, and layered tree-width.
- Need for temporalization: Static structured decompositions do not capture the temporal evolution represented by time-varying graphs.Existing temporal-graph notions differ in how they represent temporal information and changes in the underlying graph.
- Need for temporalization: The chapter develops spined structured decomposition categories for time-varying graphs and structures by reinterpreting them within persistent and cumulative narratives.This produces a time-dependent generalization of spined structured decomposition categories.
Structured decompositions.
Structured decompositions are functors from a graph-shaped indexing category into a category of objects, with every indexed morphism required to be a monomorphism.
- Definitions: A barycentric subdivision replaces each graph vertex with an object and each edge with an object mapping to both endpoints.Each subdivided edge is represented by a span between the corresponding endpoint objects.
- Examples: The barycentric subdivision of the triangle K3 illustrates how a graph shape is converted into the categorical indexing structure used for decompositions.The example is presented as a schematic visualization of the construction.
- Definitions: A J-structured decomposition in a category D is a functor from the barycentric subdivision of J to D whose indexed morphisms are monomorphisms.The monomorphism condition is the defining structural requirement stated for these decompositions.
Relevance of structured decompositions and schematic visualizations.
Structured decompositions apply across graphs, hybrid dynamical systems, groups, manifolds, and cellular sheaves, showing how a common categorical form supports diverse mathematical and temporal settings.
- Tree-decompositions: Tree-decompositions are tree-shaped structured decompositions valued in graphs, while arbitrary graph-shaped indexing yields graph-decompositions.Associated width parameters measure resemblance to models such as trees, paths, or cycles.
- Graph examples: A C5-shaped graph decomposition illustrates structured decompositions indexed by a five-cycle and its barycentric subdivision.The figure presents the cycle together with the corresponding graph-valued decomposition.
- Hybrid dynamical systems: A P3-shaped manifold decomposition uses maps among the circle, a torus, and a genus-two surface to represent a hybrid dynamical example.The map from S1 to the torus is homotopic to a constant, while the map to the genus-two surface factors through the torus.
- Graphs of groups: Applying the fundamental group functor to the manifold decomposition produces a P3-shaped structured decomposition of groups.The upper visualization shows generating curves, and the lower visualization shows the resulting group decomposition.
- Cellular sheaves: Cellular sheaves are structured co-decompositions valued in real vector spaces and are used later to model multi-agent systems with changing communication topologies.They encode local-to-global interactions in graph, simplicial-complex, and cell-complex systems.
Spined sd-categories.
Spined sd-categories extend structured decomposition categories with a graded spine of subcategories that organizes objects by structural size.
- An sd-category consists of a category D and graph class G such that every structured decomposition indexed by G admits a colimit.
- A spine is an increasing family of subcategories satisfying closure, coverage by monomorphisms, and compatibility between adjacent levels.
- A spined sd-category is the triple (D,G,Ω), combining structured decompositions with the spine that grades their constituent objects.
Combining persistent narratives and spined sd-categories.
The chapter temporalizes spined sd-categories by restricting persistent narratives to selected time intervals and transferring the static decomposition structure and spine to them.
- For a finite discrete time category, a sub-join-semilattice selects the temporal intervals used to construct persistent narratives and change temporal resolution.
- The induced subcategories bΩn contain persistent narratives whose values on selected intervals lie in the corresponding static spine level Ωn.
- Under pullback, fullness, and related assumptions on D and its spine, the persistent narrative category forms an sd-category.
- Additional pushout and pullback conditions make the induced family bΩ a spine, yielding a temporalized spined sd-category.
- This construction lifts static measures of structural complexity to persistent, time-varying data.
3.2.3 Temporalizing tree-width: examples
Temporalizing spined sd-categories produces time-varying analogues of established graph-width notions by measuring persistent narratives across selected time intervals.
- A spined sd-category assigns each object a size and extends tree-width concepts to structured decompositions and arbitrary objects.
- These examples demonstrate that ordinary, complemented, and independence-based width parameters admit natural time-varying counterparts.
- For persistent graph narratives, the temporalized size is the maximum order among graph values on the selected intervals, recovering maximum tree-width.
- The same temporalization yields maximum complemented tree-width over the selected intervals.
- A modified spine similarly recovers maximum tree independence number over the selected intervals.
3.3 Vignette 3: Temporal Cellular Sheaves: Modelling Multi-agent Systems with Switching Topologies
Temporal cellular sheaves model switching multi-agent systems by combining changing communication graphs with local state spaces, maps, and cross-time structural relationships.
- A communication graph records which agents communicate, but omits agent dynamics, exchanged information, sensing, and communication transformations.
- Cellular sheaves for multi-agent systems: Cellular sheaves assign vector spaces to agents and links, with linear maps encoding local sensing, communication, and information-processing relations.
- Cellular sheaves for multi-agent systems: Functoriality enforces compatibility between local models and graph incidences, producing a globally consistent information-processing architecture.
- Cellular sheaves for multi-agent systems: Morphisms of cellular sheaves combine an incidence-category functor with a natural transformation, representing topology changes and induced transformations of local data.
- Cellular sheaves for multi-agent systems: These morphisms support subsystem inclusion, agent aggregation into teams, and switching communication topologies.
- Temporal cellular sheaves and switching topologies: For persistent agents and links, natural transformations coherently identify local state spaces, sensing maps, and communication constraints across topology changes.
- Temporal cellular sheaves and switching topologies: Temporal cellular sheaves record how information persists and accumulates across communication architectures rather than treating switching graphs as independent snapshots.
- Temporal cellular sheaves and switching topologies: Stability theory for temporal cellular sheaves remains ongoing, including conditions for bounded and convergent target-tracking errors under topology switches.
4 Conclusion and future directions
The narrative framework unifies temporal modeling across object types, while its three research directions expose open questions about information preservation, temporalized decompositions, and switching-network control.
- The framework is largely independent of the objects evolving through time, supporting applications from categorical analysis to cellular-sheaf models of switching communication topologies.Its flexibility comes from selecting an appropriate target category for the temporal objects.
- Rigidity classes distinguish whether persistent, cumulative, or neither description survives the corresponding round trip.Rigid narratives preserve both descriptions, while left-rigid and right-rigid narratives preserve only the persistent or cumulative description, respectively.
- Future work seeks intrinsic characterizations of rigidity and its behavior under products, limits, colimits, and changes of ambient category.These questions connect categorical structure with information-preserving changes of perspective in data science.
- A width-preserving adjoint pair between persistent and cumulative temporalized spined sd-categories could transfer concepts and results between the two viewpoints.Whether the relevant adjoint functors are sd-functors or width-preserving remains an open question.
- Temporal cellular sheaves encode changing communication architectures together with heterogeneous sensing, communication, and information-processing structures in multi-agent systems.This provides a categorical foundation for distributed control on time-varying communication networks.
- A major boundary is that dynamical systems and stability theory for temporal cellular sheaves remain under development.Future work must relate dynamics to topology changes and determine when coordination objectives remain stable.