pith. machine review for the scientific record. sign in

arxiv: 2604.06800 · v1 · submitted 2026-04-08 · 🧮 math.AT

Recognition: no theorem link

A distance between maps via interleavings of relative Sullivan algebras

Katsuhiko Kuribayashi, Kengo Sekizuka, Shun Wakatsuki, Takahito Naito, Toshihiro Yamaguchi

Pith reviewed 2026-05-10 18:19 UTC · model grok-4.3

classification 🧮 math.AT
keywords relative Sullivan algebraspersistence CDGAsinterleaving distancehomotopy classes of mapsPostnikov towersdifferential graded algebrasrational homotopy theory
0
0 comments X

The pith

Relative Sullivan algebras model maps between spaces and yield persistence CDGAs whose interleavings define a pseudodistance on homotopy classes.

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

The paper associates extended tame persistence CDGAs to relative Sullivan algebras. When such an algebra models a map between spaces, the resulting persistence CDGA is isomorphic in the homotopy category to the object obtained from the map's Postnikov tower together with the polynomial de Rham functor. The authors equip this category with an interleaving distance that induces a pseudodistance directly on the homotopy set of maps. Unlike the case of persistence cochain complexes, this distance does not coincide in general with the interleaving distance on cohomology, prompting further discussion of formality questions.

Core claim

If the relative Sullivan algebra is a model for a map between spaces, then the persistence CDGA is isomorphic to the persistence object obtained by a Postnikov tower for the map with the polynomial de Rham functor in the homotopy category of extended tame persistence CDGAs.

What carries the argument

The interleaving distance in the homotopy category of extended tame persistence CDGAs, applied to persistence objects derived from relative Sullivan algebra models of maps.

If this is right

  • The pseudodistance distinguishes homotopy classes of maps that share the same cohomology but differ in higher homotopy operations.
  • It supplies an algebraic invariant that can be computed from Sullivan models and compared across different maps.
  • Formal properties of persistence CDGAs under interleavings can be studied separately from their cohomology counterparts.
  • Concrete calculations become feasible for specific maps between spaces with known rational models.

Where Pith is reading between the lines

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

  • The construction might extend to other algebraic models of maps, such as minimal models over different coefficient rings, to produce analogous distances.
  • Because the distance captures CDGA-level data rather than merely cohomology, it could separate maps whose rational homotopy types differ only in higher products or operations.
  • One could apply the distance to families of maps between spheres or Eilenberg-MacLane spaces to check whether it recovers classical homotopy invariants.

Load-bearing premise

Relative Sullivan algebras serve as valid models for maps between spaces and the interleaving distance yields a well-defined pseudodistance on the homotopy set of maps.

What would settle it

A concrete map whose relative Sullivan model produces a persistence CDGA whose interleaving distance in the homotopy category differs from the distance obtained from the corresponding Postnikov tower with the polynomial de Rham functor.

read the original abstract

In this article, we consider extended tame persistence commutative differential graded algebras (CDGAs) associated with relative Sullivan algebras. In particular, if the relative Sullivan algebra is a model for a map between spaces, then the persistence CDGA is isomorphic to the persistence object obtained by a Postnikov tower for the map with the polynomial de Rham functor in the homotopy category of extended tame persistence CDGAs. Moreover, the interleaving distance in the homotopy category (IHC) in the sense of Lanari and Scoccola enables us to introduce a pseudodistance on the homotopy set of maps via the persistence CDGA models for maps. In contrast to persistence cochain complexes, the IHC of persistence CDGAs does not coincide with the cohomology interleaving distance in general. Due to the reason, we also discuss formalities of a persistence CDGA with interleavings. Computational examples of the pseudodistances between maps are showcased.

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

2 major / 2 minor

Summary. The manuscript associates extended tame persistence CDGAs to relative Sullivan algebras modeling maps between spaces. It proves that such a persistence CDGA is isomorphic in the homotopy category of extended tame persistence CDGAs to the object obtained by applying the polynomial de Rham functor to the Postnikov tower of the map. The interleaving distance in the homotopy category (IHC) of Lanari-Scoccola is then transferred to define a pseudodistance on homotopy classes of maps. The paper notes that this IHC does not coincide with the cohomology interleaving distance, discusses formalities of persistence CDGAs under interleavings, and includes computational examples.

Significance. If the isomorphism and the transfer of the IHC hold, the construction supplies an algebraic model for a pseudodistance on maps that incorporates rational homotopy data via Sullivan algebras and persistence. This extends persistence techniques beyond spaces or functions to maps and could yield new invariants in computational homotopy theory. The explicit distinction from cohomology interleaving and the provision of examples are positive features that strengthen the contribution.

major comments (2)
  1. [Formalities section (discussion of persistence CDGAs with interleavings)] The central claim that the IHC yields a pseudodistance on homotopy sets of maps rests on the well-definedness of the interleaving distance in the homotopy category of extended tame persistence CDGAs. Because the manuscript explicitly states that this IHC does not coincide with the cohomology interleaving distance, the formalities discussion must independently verify non-negativity, symmetry, the triangle inequality, and d(x,x)=0 after passage to homotopy; this verification is load-bearing and should be expanded with explicit arguments rather than relying on the underlying CDGA category alone.
  2. [Main construction (isomorphism claim)] The isomorphism statement between the persistence CDGA coming from the relative Sullivan algebra and the Postnikov-tower construction via the polynomial de Rham functor is asserted in the homotopy category; the precise functoriality, the role of the homotopy category of extended tame persistence CDGAs, and the compatibility with the model structure must be stated with reference to the relevant diagrams or propositions to guarantee that the distance is independent of the choice of model.
minor comments (2)
  1. Notation for 'extended tame persistence CDGAs' and the precise meaning of 'homotopy category' in this context should be fixed at the first appearance to aid readability.
  2. The computational examples would be strengthened by including the explicit relative Sullivan algebras or the spaces/maps used, together with the resulting persistence CDGAs, to allow independent verification of the reported pseudodistances.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We appreciate the recognition of the potential significance of our construction in extending persistence techniques to maps via rational homotopy theory. Below, we address each major comment point by point, indicating the revisions we have made to the manuscript.

read point-by-point responses

  1. Referee: [Formalities section (discussion of persistence CDGAs with interleavings)] The central claim that the IHC yields a pseudodistance on homotopy sets of maps rests on the well-definedness of the interleaving distance in the homotopy category of extended tame persistence CDGAs. Because the manuscript explicitly states that this IHC does not coincide with the cohomology interleaving distance, the formalities discussion must independently verify non-negativity, symmetry, the triangle inequality, and d(x,x)=0 after passage to homotopy; this verification is load-bearing and should be expanded with explicit arguments rather than relying on the underlying CDGA category alone.

    Authors: We agree that an expanded discussion is warranted. In the revised version, we have added explicit arguments verifying that the interleaving distance descends to a pseudodistance on the homotopy category of extended tame persistence CDGAs. These include direct checks for non-negativity, symmetry, the triangle inequality, and reflexivity (d(x,x)=0), leveraging the fact that weak equivalences in the model structure preserve interleavings up to homotopy. This addresses the concern that the distance does not coincide with the cohomology interleaving distance, ensuring the properties hold independently. revision: yes

  2. Referee: [Main construction (isomorphism claim)] The isomorphism statement between the persistence CDGA coming from the relative Sullivan algebra and the Postnikov-tower construction via the polynomial de Rham functor is asserted in the homotopy category; the precise functoriality, the role of the homotopy category of extended tame persistence CDGAs, and the compatibility with the model structure must be stated with reference to the relevant diagrams or propositions to guarantee that the distance is independent of the choice of model.

    Authors: We have revised the main construction section to include a more detailed description of the functoriality. We now explicitly reference the propositions establishing the model structure on extended tame persistence CDGAs and the compatibility of the relative Sullivan algebra construction with the polynomial de Rham functor applied to the Postnikov tower. A diagram showing the natural transformation between the two constructions is included, demonstrating that the isomorphism in the homotopy category is natural. This ensures that the induced interleaving distance is independent of the choice of model for the map, as required. revision: yes

Circularity Check

0 steps flagged

No circularity detected in derivation of pseudodistance via persistence CDGAs

full rationale

The paper defines extended tame persistence CDGAs from relative Sullivan algebra models of maps, proves an isomorphism in the homotopy category to the Postnikov tower construction via the polynomial de Rham functor, and transfers the interleaving distance (IHC) from the external reference Lanari-Scoccola to obtain a pseudodistance on homotopy classes of maps. It explicitly discusses formalities to establish the pseudodistance axioms on this category and notes that IHC on CDGAs differs from the cohomology version, so the central claim rests on independent verification rather than any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The construction is self-contained against the cited external notions and benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard background from algebraic topology and persistence theory; no free parameters are introduced in the abstract, and the new objects are extensions rather than invented entities with independent evidence.

axioms (2)
  • domain assumption Relative Sullivan algebras model maps between spaces
    Invoked in the statement that the persistence CDGA corresponds to the map model.
  • domain assumption The homotopy category of extended tame persistence CDGAs admits a well-defined interleaving distance
    Central to defining the pseudodistance on maps.

pith-pipeline@v0.9.0 · 5472 in / 1287 out tokens · 46148 ms · 2026-05-10T18:19:42.075785+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

34 extracted references · 6 canonical work pages · 1 internal anchor

  1. [1]

    Andersen and J

    K.K.S. Andersen and J. Grodal, A Baues fibration category structure on Banach andC ∗- algebras, preprint, available athttps://web.math.ku.dk/~jg/papers/fibcat.pdf

  2. [2]

    H. J. Baues, Algebraic homotopy, Cambridge studies in advanced mathematics 15, Cambridge University Press, 1989

    1989

  3. [3]

    Buijs, F

    U. Buijs, F. F´ elix, A. Murillo and D. Tanr´ e, Lie models in topology, Progress in Mathematics 335, Cham: Birkh¨ auser, 2020

    2020

  4. [4]

    Blumberg and M

    A.J. Blumberg and M. Lesnick, Universality of the Homotopy Interleaving Distance, Trans. Amer. Math. Soc.376(2023), 8269–8307. DOI:https://doi.org/10.1090/tran/8738

    work page doi:10.1090/tran/8738 2023

  5. [5]

    Bousfield and V.K.A.M

    A.K. Bousfield and V.K.A.M. Gugenheim, On PL de Rham theory and rational homotopy type, Memoirs of AMS179(1976)

    1976

  6. [6]

    Brian, O

    P. Brian, O. Cornea and J. Zhang, Triangulation, Persistence, and Fukaya categories, to appear in Memoirs of the European Mathematical Society

  7. [7]

    Bubenik and J.A

    P. Bubenik and J.A. Scott, Categorification of Persistent Homology, Discrete Comput. Geom. 51(2014), 600–627

    2014

  8. [8]

    Carranza, B

    D. Carranza, B. Doherty, K. Kapulkin, M. Opie, M. Sarazola and L.Z. Wong, Cofibration category of digraphs for path homology, Algebr. Comb.7(2024), 475–514

    2024

  9. [9]

    Chach´ olski, B

    W. Chach´ olski, B. Giunti and C. Landi, Invariants for tame parametrised chain complexes, Homology, Homotopy and Applications,23(2021), 183–213

    2021

  10. [10]

    Chazal, D

    F. Chazal, D. Cohen-Steiner, M. Glisse, L.J. Guibas and S.Y. Oudot, Proximity of persis- tence modules and their diagrams, In Proceedings of the twenty-fifth annual symposium on Computational geometry (2009), 237–246

    2009

  11. [11]

    Dwyer and J

    W.G. Dwyer and J. Spali´ nski, Homotopy theories and model categories. In Handbook of algebraic topology, pages 73–126. North-Holland, Amsterdam, 1995

    1995

  12. [12]

    F´ elix and S

    Y. F´ elix and S. Halperin, Formal sapce with finite-dimensional rational homotopy, Trans.A.M.S bf 270 (1982) 575-588

    1982

  13. [13]

    F´ elix, M

    Y. F´ elix, M. Fuentes and A. Murillo, A Lie characterization of te Bousfield-KanQ-completion andQ-good space, preprint, available atarXiv:2407.02812v1

    work page arXiv

  14. [14]

    F´ elix, S

    Y. F´ elix, S. Halperin and J.-C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics205, Springer-Verlag, 2000

    2000

  15. [15]

    F´ elix, S

    Y. F´ elix, S. Halperin and J.-C. Thomas, Rational homotopy theory II, World Scientific, 2015

    2015

  16. [16]

    F´ elix, J

    Y. F´ elix, J. Oprea and D. Tanr´ e, Algebraic Models in Geometry, Oxford Graduate Texts in Mathematics, Oxford University Press, 2008

    2008

  17. [17]

    Griffiths and J

    P. Griffiths and J. Morgan, Rational Homotopy Theory and Differential Forms, Birkh¨ auser 16, 1981

    1981

  18. [18]

    P. G. Goerss and J. F. Jardine, Simplicial homotopy theory. Springer Science & Business Media, 2009

    2009

  19. [19]

    Halperin, Lectures on minimal models, M´ emoires S.M.F

    S. Halperin, Lectures on minimal models, M´ emoires S.M.F. Nouvelle s´ erie9-10, 1983

    1983

  20. [20]

    K. Hess, S. Lavenir and K. Maggs, Cell decompositions of persistent minimal models, 2024, preprint, available athttps://arxiv.org/abs/2312.08326v2

    work page arXiv 2024

  21. [21]

    Halperin and J

    S. Halperin and J. Stasheff, Obstructions to homotopy equivalences, Advances in Mathematics 32(1979) 233-279

    1979

  22. [22]

    Kuribayashi, On F´ elix and Tanr´ e rational models for polyhedral products, Fundamenta Mathematicae267(2024), 243–265

    K. Kuribayashi, On F´ elix and Tanr´ e rational models for polyhedral products, Fundamenta Mathematicae267(2024), 243–265

    2024

  23. [23]

    Kuribayashi, T

    K. Kuribayashi, T. Naito, S. Wakatsuki and T. Yamaguchi, Algebraic interleavings of spaces over the classifying space of the circle, 2025, preprint,https://arxiv:2501.09257. The equal- ities of interleaving distances and cohomology interleavings of spaces overBS 1 (the revised version), 2025. 40 K. KURIBAYASHI, T. NAITO, K. SEKIZUKA, S. WAKATSUKI, AND T. ...

    work page arXiv 2025

  24. [24]

    Lanari and L

    E. Lanari and L. Scoccola, Rectification of interleavings and a persistent Whitehead theorem Algebraic & Geometric Topology23(2023), 803–832

    2023

  25. [25]

    J. P. May, A concise course in algebraic topology. University of Chicago press, 1999

    1999

  26. [26]

    M´ emoli and L

    F. M´ emoli and L. Zhou, Persistent homotopy groups of metric spaces, Journal of Topology and Analysis17(2025), 1481–1542

    2025

  27. [27]

    Mohamad and P.D

    N.E.G. Mohamad and P.D. Mitchner, Coarse examples of Baues cofibration category, SARA- JEVO JOURNAL OF MATHEMATICS,16(2020), 83–103

    2020

  28. [28]

    Algebraic models for equivariant rational homotopy theory for discrete groups

    J.M. Moreno-Fern´ andez and B. Stonek, Algebraic models for equivariant rational homotopy theory for discrete groups, preprintarXiv:2601.17345v1

    work page internal anchor Pith review Pith/arXiv arXiv

  29. [29]

    Riehl and D

    E. Riehl and D. Verity, Elements of∞-category theory, Cambridge Studies in Advanced Mathematics 194. Cambridge: Cambridge University Press, 2022

    2022

  30. [30]

    L. N. Scoccola, Locally persistent categories and metric properties of interleaving distances, 2020, Thesis, The University of Western Ontario

    2020

  31. [31]

    Sekizuka, Extended tame persistence objects with values in a cofibration category, in preparation

    K. Sekizuka, Extended tame persistence objects with values in a cofibration category, in preparation

  32. [32]

    Thomas, Rational homotopy of Serre fibrations, Ann.Inst.Fourier, Grenoble31(1981) 71-90

    J.-C. Thomas, Rational homotopy of Serre fibrations, Ann.Inst.Fourier, Grenoble31(1981) 71-90

    1981

  33. [33]

    Thomas, Eilenberg–Moore models for fibrations, Trans

    J.-C. Thomas, Eilenberg–Moore models for fibrations, Trans. Amer. Math. Soc.274(1982), 203–225

    1982

  34. [34]

    Ling Zhou, Persistent Sullivan minimal models of metric spaces, 2023, preprint, available at https://arxiv.org/abs/2310.06263. Department of Mathematics, F aculty of Science, Shinshu University, Matsumoto, Nagano 390-8621, Japan Email address:kuri@math.shinshu-u.ac.jp Nippon Institute of Technology, Gakuendai, Miyashiro-machi, Minamisaitama-gun, Saitama 3...

    work page arXiv 2023