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arxiv: 2402.00206 · v3 · submitted 2024-01-31 · 🧮 math.CT · cs.DS

Towards a Unified Theory of Time-Varying Data

Benjamin Merlin Bumpus , James Fairbanks , Martti Karvonen , Wilmer Leal , Fr\'ed\'eric Simard This is my paper

Pith reviewed 2026-05-24 03:30 UTC · model grok-4.3

classification 🧮 math.CT cs.DS
keywords time-varying graphssheavesnarrativescategory theorytemporal dataposetsadjunctionspersistent interpretations
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The pith

Sheaves on posets of time intervals define categories of narratives that unify time-varying data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces categories of narratives as sheaves on posets of time intervals to encode snapshots of temporal objects and their relationships. This construction satisfies five desiderata from time-varying graph theory: defining objects and morphisms, distinguishing cumulative and persistent interpretations with principled transitions, lifting static notions temporally, being object-agnostic, and integrating with dynamical systems. By generalizing from existing approaches to any category with limits and colimits, it formalizes tacit intuitions and proves new results like the adjunction between persistent and cumulative narratives. A sympathetic reader would care because it offers a consistent framework for handling evolving mathematical structures without ad-hoc definitions.

Core claim

We introduce 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 and generalizes to any category with limits and colimits, including an adjunction between persistent and cumulative narratives.

What carries the argument

Categories of narratives, defined as sheaves on posets of time intervals that encode snapshots and relationships of temporal objects.

If this is right

  • It defines both time-varying objects and their morphisms.
  • It distinguishes between cumulative and persistent interpretations and provides principled methods for transitioning between them.
  • It systematically lifts static notions to their temporal analogues.
  • It is object agnostic.
  • It integrates with theories of dynamical systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach could extend to modeling time-varying phenomena in fields like biology or physics by applying the same sheaf construction.
  • The adjunction between persistent and cumulative narratives may allow for efficient switching between data representations in computational tools.
  • Generalizing beyond graphs suggests the framework applies to time-varying topological spaces or other structures without modification.
  • Future work might test this on real datasets to see if it captures intuitions better than existing ad-hoc methods.

Load-bearing premise

That the sheaf construction on posets of time intervals will capture the tacit intuitions of temporal graph theory and provide principled transitions between cumulative and persistent interpretations without additional ad-hoc choices when generalized to categories with limits and colimits.

What would settle it

A counterexample time-varying graph where the narrative sheaf either fails to distinguish cumulative from persistent interpretations or requires ad-hoc choices to match established temporal graph definitions.

Figures

Figures reproduced from arXiv: 2402.00206 by Benjamin Merlin Bumpus, Fr\'ed\'eric Simard, James Fairbanks, Martti Karvonen, Wilmer Leal.

Figure 1
Figure 1. Figure 1: A schematic visualization of a sheaf on a discrete time category (a persistent narrative) with three [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A temporal graph along with its persistent and cumulative narratives [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper introduces categories of narratives, defined as sheaves on posets of time intervals valued in any category with limits and colimits. These encode snapshots of temporal objects and their relationships. The framework is claimed to satisfy five desiderata from time-varying graph theory: (D1) defining both objects and morphisms, (D2) distinguishing cumulative and persistent interpretations with a principled adjunction-based transition, (D3) systematically lifting static notions to temporal analogues, (D4) being object-agnostic, and (D5) integrating with dynamical systems. It generalizes prior categorical/sheaf approaches, formalizes implicit intuitions, and introduces novel elements such as the adjunction between persistent and cumulative narratives.

Significance. If the constructions, proofs, and generalization hold as stated, the work offers a coherent categorical synthesis that unifies disparate ideas in temporal data and provides a principled mechanism (via adjunction) for switching interpretations. The object-agnostic scope and integration with dynamical systems could enable broader applications beyond graphs. The explicit introduction of the adjunction as a new result in the literature is a concrete strength.

minor comments (2)
  1. Abstract: the five desiderata are referenced but not enumerated; a one-sentence listing of D1–D5 would improve immediate readability for readers unfamiliar with the temporal-graph literature.
  2. The claim that the adjunction 'has not appeared in the temporal data literature' is presented without a supporting citation or literature survey section; adding a brief related-work paragraph would strengthen the novelty statement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their detailed summary of the manuscript and for recognizing its significance as a coherent categorical synthesis that unifies ideas in temporal data, along with the explicit introduction of the adjunction between persistent and cumulative narratives. The recommendation for minor revision is noted. As the report lists no specific major comments, we have no points requiring point-by-point response or revision.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines categories of narratives as sheaves on posets of time intervals valued in categories with limits and colimits, then verifies that this construction satisfies five listed desiderata (D1-D5) via standard sheaf and adjunction machinery. The adjunction between persistent and cumulative narratives is explicitly presented as a novel result not previously appearing in the temporal data literature. No equations or definitions reduce by construction to fitted parameters, renamed inputs, or self-citations whose content is itself unverified; the framework is scoped to external categorical foundations and adds independent formal content. The derivation chain is therefore self-contained against the benchmarks of sheaf theory and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The framework rests on the standard axioms of category theory and sheaf theory together with the new definition of narratives; no numerical free parameters are introduced. The central invented entity is the category of narratives itself.

axioms (1)
  • standard math Standard axioms of category theory (objects, morphisms, composition, identities) and sheaf theory on posets.
    The paper states it builds upon existing categorical and sheaf-theoretic approaches to temporal graph theory.
invented entities (1)
  • categories of narratives no independent evidence
    purpose: To encode snapshots of temporal objects and relationships between them as sheaves on posets of time intervals, satisfying the five desiderata.
    New definition introduced to generalize static notions to temporal analogues in an object-agnostic way.

pith-pipeline@v0.9.0 · 5794 in / 1426 out tokens · 50997 ms · 2026-05-24T03:30:50.896464+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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  1. Maximizing Reachability via Shifting of Temporal Paths

    cs.DS 2026-05 unverdicted novelty 6.0

    Maximizing reachability in k-path temporal graphs via budgeted shifts is FPT when parameterized by k and b together or by k alone, but intractable in most other parameterizations with matching XP algorithms.

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    S., P ATTERSON , E., AND SHAPIRO , B

    A DUDDELL , R., F AIRBANKS , J., K UMAR , A., O CAL , P. S., P ATTERSON , E., AND SHAPIRO , B. T. A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks. arXiv preprint arXiv:2301.01445 (2023)

    work page arXiv 2023

  2. [2]

    Confessions, volume ii: Books 9–13

    A UGUSTINE , S. Confessions, volume ii: Books 9–13. edited and translated by c j.-b. hammond. loeb classical library 27, isbn 0-67499693-3, 2016

    work page 2016

  3. [3]

    Category theory

    A WODEY, S. Category theory. Oxford University Press, 2010. ISBN:0199237182

    work page 2010

  4. [4]

    Group formation in large social networks: Membership, growth, and evolution

    B ACKSTROM , L., H UTTENLOCHER , D., K LEINBERG , J., AND LAN, X. Group formation in large social networks: Membership, growth, and evolution. In Proceedings of the 12th ACM SIGKDD Inter- national Conference on Knowledge Discovery and Data Mining (New York, NY , USA, 2006), KDD ’06, Association for Computing Machinery, p. 44–54

    work page 2006

  5. [5]

    On the enumeration of maximal ( δ, γ)-cliques of a temporal network

    B ANERJEE , S., AND PAL, B. On the enumeration of maximal ( δ, γ)-cliques of a temporal network. In Proceedings of the ACM India Joint International Conference on Data Science and Management of Data (2019), pp. 112–120

    work page 2019

  6. [6]

    Listing all maximal k-plexes in temporal graphs

    B ENTERT , M., H IMMEL , A.-S., M OLTER , H., M ORIK , M., N IEDERMEIER , R., AND SAITENMACHER , R. Listing all maximal k-plexes in temporal graphs. Journal of Experimental Algorithmics (JEA) 24 (2019), 1–27

    work page 2019

  7. [7]

    B UMPUS , B. M. Generalizing graph decompositions. PhD thesis, University of Glasgow, 2021

    work page 2021

  8. [8]

    M., AND MEEKS , K

    B UMPUS , B. M., AND MEEKS , K. Edge exploration of temporal graphs. Algorithmica (2022), 1–29

    work page 2022

  9. [9]

    Time-varying graphs and dynamic networks

    C ASTEIGTS , A., F LOCCHINI , P., QUATTROCIOCCHI , W., AND SANTORO , N. Time-varying graphs and dynamic networks. International Journal of Parallel, Emergent and Distributed Systems 27 , 5 (2012), 387–408

    work page 2012

  10. [10]

    Sheaves, Cosheaves and Applications

    C URRY, J. Sheaves, Cosheaves and Applications. PhD thesis, University of Pennsylvania, 2014

    work page 2014

  11. [11]

    C URRY, J. M. Topological data analysis and cosheaves. Japan Journal of Industrial and Applied Mathe- matics 32, 2 (Jul 2015), 333–371

    work page 2015

  12. [12]

    Categorified reeb graphs

    DE SILVA, V., M UNCH , E., AND PATEL, A. Categorified reeb graphs. Discrete & Computational Geometry 55, 4 (Jun 2016), 854–906

    work page 2016

  13. [13]

    E NRIGHT , J., AND KAO, R. R. Epidemics on dynamic networks. Epidemics 24 (2018), 88–97

    work page 2018

  14. [14]

    B., AND ZAMARAEV , V

    E NRIGHT , J., M EEKS , K., M ERTZIOS , G. B., AND ZAMARAEV , V. Deleting edges to restrict the size of an epidemic in temporal networks. Journal of Computer and System Sciences 119 (2021), 60–77

    work page 2021

  15. [15]

    Assigning times to minimise reachability in temporal graphs

    E NRIGHT , J., M EEKS , K., AND SKERMAN , F. Assigning times to minimise reachability in temporal graphs. Journal of Computer and System Sciences 115 (2021), 169–186

    work page 2021

  16. [16]

    F ONG , B., AND SPIVAK , D. I. An Invitation to Applied Category Theory: Seven Sketches in Composi- tionality. Cambridge University Press, 2019. 23

    work page 2019

  17. [17]

    Dynamic graph models

    H ARARY, F., AND GUPTA, G. Dynamic graph models. Mathematical and Computer Modelling 25 , 7 (1997), 79–87

    work page 1997

  18. [18]

    Temporal interval cliques and independent sets

    H ERMELIN , D., I TZHAKI , Y., M OLTER , H., AND NIEDERMEIER , R. Temporal interval cliques and independent sets. Theoretical Computer Science (2023), 113885

    work page 2023

  19. [19]

    Adapting the bron–kerbosch algorithm for enumerating maximal cliques in temporal graphs

    H IMMEL , A.-S., M OLTER , H., N IEDERMEIER , R., AND SORGE , M. Adapting the bron–kerbosch algorithm for enumerating maximal cliques in temporal graphs. Social Network Analysis and Mining 7 (2017), 1–16

    work page 2017

  20. [20]

    Modern temporal network theory: a colloquium

    H OLME , P. Modern temporal network theory: a colloquium. The European Physical Journal B 88 , 9 (2015), 1–30

    work page 2015

  21. [21]

    Temporal networks

    H OLME , P., AND SARAMÄKI , J. Temporal networks. Physics Reports 519, 3 (2012), 97–125

    work page 2012

  22. [22]

    Protocols and impossibility results for gossip-based communication mechanisms

    K EMPE , D., AND KLEINBERG , J. Protocols and impossibility results for gossip-based communication mechanisms. In The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceed- ings. (2002), pp. 471–480

    work page 2002

  23. [23]

    Connectivity and inference problems for temporal networks

    K EMPE , D., K LEINBERG , J., AND KUMAR , A. Connectivity and inference problems for temporal networks. Journal of Computer and System Sciences 64, 4 (2002), 820–842

    work page 2002

  24. [24]

    Maximizing the spread of influence through a social net- work

    K EMPE , D., K LEINBERG , J., AND TARDOS , E. Maximizing the spread of influence through a social net- work. In Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (New York, NY , USA, 2003), KDD ’03, Association for Computing Machinery, p. 137–146

    work page 2003

  25. [25]

    Extracting persistent clusters in dynamic data via möbius inversion.Discrete & Computational Geometry (Oct 2023)

    K IM, W., AND MÉMOLI , F. Extracting persistent clusters in dynamic data via möbius inversion.Discrete & Computational Geometry (Oct 2023)

    work page 2023

  26. [26]

    Adhesive categories

    L ACK, S., AND SOBOCINSKI , P. Adhesive categories. In Foundations of Software Science and Computa- tion Structures (Berlin, Heidelberg, 2004), I. Walukiewicz, Ed., Springer Berlin Heidelberg, pp. 273–288

    work page 2004

  27. [27]

    The Experience and Perception of Time

    L E POIDEVIN , R. The Experience and Perception of Time. In The Stanford Encyclopedia of Philosophy, E. N. Zalta, Ed., Summer 2019 ed. Metaphysics Research Lab, Stanford University, 2019

    work page 2019

  28. [28]

    Exploration of Chemical Space: Formal, chemical and historical aspects

    L EAL , W. Exploration of Chemical Space: Formal, chemical and historical aspects. PhD thesis, Disser- tation, Leipzig, Universität Leipzig, 2022, 2022

    work page 2022

  29. [29]

    J., L EAL , W., L UU, D

    L LANOS , E. J., L EAL , W., L UU, D. H., J OST, J., S TADLER , P. F., AND RESTREPO , G. Exploration of the chemical space and its three historical regimes. Proceedings of the National Academy of Sciences 116, 26 (2019), 12660–12665

    work page 2019

  30. [30]

    Sheaves in geometry and logic: A first introduction to topos theory

    M ACLANE , S., AND MOERDIJK , I. Sheaves in geometry and logic: A first introduction to topos theory. Springer Science & Business Media, 2012

    work page 2012

  31. [31]

    The Open Algebraic Path Problem

    M ASTER , J. The Open Algebraic Path Problem. In LIPIcs Proceedings of CALCO 2021 (2021), Schloss Dagstuhl, pp. 20:1–20:20

    work page 2021

  32. [32]

    An introduction to temporal graphs: An algorithmic perspective

    M ICHAIL , O. An introduction to temporal graphs: An algorithmic perspective. Internet Mathematics 12, 4 (2016), 239–280

    work page 2016

  33. [33]

    Isolation concepts applied to temporal clique enumeration

    M OLTER , H., N IEDERMEIER , R., AND RENKEN , M. Isolation concepts applied to temporal clique enumeration. Network Science 9, S1 (2021), S83–S105

    work page 2021

  34. [34]

    Confessions, Volume I: Books 1–8, vol

    OF HIPPO , A. Confessions, Volume I: Books 1–8, vol. 26 of Loeb Classical Library. Harvard University Press, Cambridge, MA, 1912. 24

    work page 1912

  35. [35]

    Categorical Data Structures for Technical Comput- ing

    P ATTERSON , E., L YNCH , O., AND FAIRBANKS , J. Categorical Data Structures for Technical Comput- ing. Compositionality 4 (Dec. 2022)

    work page 2022

  36. [36]

    Category theory in context

    R IEHL , E. Category theory in context. Courier Dover Publications, 2017. ISBN:048680903X

    work page 2017

  37. [37]

    Sheaf Theory through Examples

    R OSIAK , D. Sheaf Theory through Examples. The MIT Press, 10 2022

    work page 2022

  38. [38]

    T., B EAUNÉE , G., B ANKS , C

    R UGET , A.-S., R OSSI , G., P EPLER , P. T., B EAUNÉE , G., B ANKS , C. J., E NRIGHT , J., AND KAO, R. R. Multi-species temporal network of livestock movements for disease spread. Applied Network Science 6, 1 (2021), 1–20

    work page 2021

  39. [39]

    S CHULTZ , P., AND SPIVAK , D. I. Temporal type theory: A topos-theoretic approach to systems and behavior. arXiv preprint arXiv:1710.10258 (2017)

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  40. [40]

    I., AND VASILAKOPOULOU , C

    S CHULTZ , P., S PIVAK , D. I., AND VASILAKOPOULOU , C. Dynamical systems and sheaves. Applied Categorical Structures 28, 1 (2020), 1–57

    work page 2020

  41. [41]

    Identifying roles in an ip network with temporal and structural density

    V IARD , J., AND LATAPY, M. Identifying roles in an ip network with temporal and structural density. In 2014 IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS) (2014), IEEE, pp. 801–806

    work page 2014

  42. [42]

    Computing maximal cliques in link streams

    V IARD , T., L ATAPY, M., AND MAGNIEN , C. Computing maximal cliques in link streams. Theoretical Computer Science 609 (2016), 245–252. 25

    work page 2016