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arxiv: 2606.09747 · v1 · pith:H46DSAEWnew · submitted 2026-06-08 · 🧮 math.AT · math.CT

Homotopy theories via the magnitude-path spectral sequence

Muriel Livernet , Emily Roff , Sarah Whitehouse This is my paper

Pith reviewed 2026-06-27 14:06 UTC · model grok-4.3

classification 🧮 math.AT math.CT
keywords magnitude homologypath homologyspectral sequencehomotopy theoryEilenberg-Steenrod axiomsmetric spacesBrown categoriesdirected graphs
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The pith

The magnitude-path spectral sequence equips each page with a metric version of the Eilenberg-Steenrod axioms on generalized metric spaces.

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

The paper introduces a family of homotopy theories on generalized metric spaces with natural number distances, built from the magnitude-path spectral sequence. The first page recovers magnitude homology and the second contains bigraded path homology. For each natural number r the authors define r-quasi-isomorphisms and r-cofibrations so that page r satisfies the Eilenberg-Steenrod axioms in a metric sense, including a Mayer-Vietoris theorem. Brown category structures are then placed on the spaces, making homotopy colimits explicit and allowing concrete descriptions of r-suspensions and r-spheres; the case r equals 1 restricts the whole theory to directed graphs.

Core claim

The magnitude-path spectral sequence is a filtered chain complex whose r-th page carries a homology theory on generalized metric spaces once r-quasi-isomorphisms and r-cofibrations are introduced; each such page satisfies metric analogues of the Eilenberg-Steenrod axioms, admits a Mayer-Vietoris theorem, and participates in a Brown category structure that computes homotopy colimits, with explicit computations available for r-suspensions, r-spheres, and the restriction to directed graphs at r=1.

What carries the argument

The magnitude-path spectral sequence, whose pages are turned into homology theories by the auxiliary classes of r-quasi-isomorphisms and r-cofibrations.

If this is right

  • Page r of the spectral sequence defines a homology theory satisfying excision and Mayer-Vietoris with respect to r-cofibrations.
  • The Brown category structures permit explicit computation of all homotopy colimits in the category of generalized metric spaces.
  • r-suspensions and r-spheres of any dimension n admit concrete descriptions whose spectral sequences can be computed directly.
  • When r equals 1 the entire construction restricts to a homotopy theory on the category of directed graphs.

Where Pith is reading between the lines

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

  • The different pages may supply a sequence of successively finer metric homotopy invariants that interpolate between magnitude homology and path homology.
  • The restriction to natural-number distances suggests that the theories could be extended to arbitrary metric spaces by taking limits or approximations.
  • The Brown-category formalism might be portable to other filtered chain complexes arising in geometric or combinatorial settings.

Load-bearing premise

The magnitude-path spectral sequence is well-defined on generalized metric spaces with natural number distances and the auxiliary notions of r-quasi-isomorphism and r-cofibration make each page satisfy the Eilenberg-Steenrod axioms.

What would settle it

A generalized metric space together with an r-cofibration such that the Mayer-Vietoris sequence fails to be exact at page r of the spectral sequence.

read the original abstract

We introduce a family of homotopy theories for generalized metric spaces with natural number distances, via the magnitude-path spectral sequence (MPSS). The first page of the MPSS is known as magnitude homology; the second page is known as bigraded path homology, and contains GLMY path homology as its top row. For each natural number r, we define a class of maps of metric spaces called r-quasi-isomorphisms: those maps that induce a quasi-isomorphism at page r of the MPSS. We show that every page of the spectral sequence satisfies a suitable metric analogue of each of the Eilenberg-Steenrod axioms. In particular, we introduce the notion of r-cofibration and prove a Mayer-Vietoris theorem for page r with respect to r-cofibrations. We establish a family of Brown category structures on generalized metric spaces which allow us to explicitly compute homotopy colimits. We apply this to describe r-suspension and r-spheres of dimension n, and compute their spectral sequences. Finally we prove that for r = 1 the entire theory restricts to directed graphs.

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 / 1 minor

Summary. The manuscript introduces the magnitude-path spectral sequence (MPSS) for generalized metric spaces with natural number distances. The first page recovers magnitude homology and the second page recovers bigraded path homology (with GLMY path homology as its top row). For each r it defines r-quasi-isomorphisms and r-cofibrations, claims that every page satisfies a metric analogue of the Eilenberg-Steenrod axioms (including a Mayer-Vietoris theorem), constructs Brown category structures to compute homotopy colimits, describes r-suspensions and r-spheres together with their spectral sequences, and shows that the r=1 case restricts to directed graphs.

Significance. If the constructions and proofs are correct, the work supplies a coherent family of homotopy theories that interpolates between magnitude homology and path homology and equips metric spaces with computable homotopy colimits via Brown categories. The explicit treatment of r-spheres and the consistency check for directed graphs are concrete strengths.

major comments (2)
  1. [Opening paragraphs / definition of MPSS] The definition and well-definedness of the MPSS itself (including the filtration that produces the spectral sequence) is load-bearing for every subsequent claim; without an explicit construction it is impossible to verify that r-quasi-isomorphisms are well-defined or that the pages satisfy the metric Eilenberg-Steenrod axioms.
  2. [Mayer-Vietoris theorem] The proof of the Mayer-Vietoris theorem for page r with respect to r-cofibrations is central to the claim that each page is a homology theory; the argument must be checked for any implicit restrictions on the metric or on the supports of the chains that would prevent the long exact sequence from holding in full generality.
minor comments (1)
  1. [Abstract / introduction] The relationship between the second page of the MPSS and GLMY path homology should be stated with a precise reference to the row in which the latter appears.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their report and for highlighting the foundational aspects of the magnitude-path spectral sequence. We address each major comment below with references to the explicit constructions in the manuscript.

read point-by-point responses

  1. Referee: [Opening paragraphs / definition of MPSS] The definition and well-definedness of the MPSS itself (including the filtration that produces the spectral sequence) is load-bearing for every subsequent claim; without an explicit construction it is impossible to verify that r-quasi-isomorphisms are well-defined or that the pages satisfy the metric Eilenberg-Steenrod axioms.

    Authors: Section 2 of the manuscript gives the explicit definition of the MPSS. The underlying chain complex is the path complex on the generalized metric space (with natural-number distances), filtered by a parameter-dependent filtration that isolates the r-component; the spectral sequence is the one associated to this filtered complex, with differentials the alternating sums of face maps. r-quasi-isomorphisms are defined directly as maps inducing isomorphisms on the r-th page. The well-definedness of the pages and the induced maps follows from the standard convergence and naturality properties of spectral sequences of filtered chain complexes, which are verified before the Eilenberg-Steenrod axioms are checked in later sections. revision: no

  2. Referee: [Mayer-Vietoris theorem] The proof of the Mayer-Vietoris theorem for page r with respect to r-cofibrations is central to the claim that each page is a homology theory; the argument must be checked for any implicit restrictions on the metric or on the supports of the chains that would prevent the long exact sequence from holding in full generality.

    Authors: Section 4 contains the proof of the Mayer-Vietoris theorem for each page r relative to r-cofibrations. The argument applies verbatim to all generalized metric spaces with natural-number distances: the chain groups consist of all finite paths whose lengths satisfy the metric inequalities, with no further restrictions on supports. The r-cofibration induces a short exact sequence of filtered chain complexes whose associated long exact sequence in homology holds on every page; the proof uses only the exactness of the sequence of chain groups and the compatibility of the filtration with the boundary maps. revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines the magnitude-path spectral sequence and new classes (r-quasi-isomorphisms, r-cofibrations) from first principles on generalized metric spaces, then proves that each page satisfies metric Eilenberg-Steenrod axioms including a Mayer-Vietoris theorem. These steps are presented as independent constructions without any quoted reduction of a central claim to a fitted parameter, self-citation chain, or definitional equivalence. The abstract and described results contain no load-bearing self-referential steps of the enumerated kinds.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claims rest on the existence and properties of the magnitude-path spectral sequence together with several new definitions introduced in the paper. No numerical free parameters are mentioned. The work relies on standard homological algebra and category theory plus domain-specific assumptions about metric spaces.

axioms (2)
  • standard math Spectral sequences exist and converge in the homological algebra setting used for the MPSS.
    Invoked to define the pages and their properties.
  • domain assumption Generalized metric spaces with natural number distances form a category in which the MPSS and the stated axioms can be formulated.
    The entire theory is built on this class of spaces.
invented entities (2)
  • r-quasi-isomorphism no independent evidence
    purpose: Class of maps that induce quasi-isomorphisms at page r of the MPSS, used to define the homotopy theory.
    New definition introduced to equip each page with a notion of equivalence.
  • r-cofibration no independent evidence
    purpose: Class of maps used to state and prove the Mayer-Vietoris theorem at page r.
    Invented to make the Eilenberg-Steenrod axioms hold in the metric setting.

pith-pipeline@v0.9.1-grok · 5717 in / 1597 out tokens · 33026 ms · 2026-06-27T14:06:46.729127+00:00 · methodology

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Reference graph

Works this paper leans on

33 extracted references · 7 canonical work pages · 4 internal anchors

  1. [1]

    Magnitude homology of geodesic metric spaces with an upper curvature bound

    Yasuhiko Asao. Magnitude homology of geodesic metric spaces with an upper curvature bound. Algebr. Geom. Topol., 21(2):647–664, 2021

    2021

  2. [2]

    Asao,Magnitude and magnitude homology of filtered set enriched categories, (2023), arXiv:2303.05677

    Yasuhiko Asao. Magnitude and magnitude homology of filtered set enriched categories. arXiv:2303.05677, 2023

    work page arXiv 2023

  3. [3]

    Magnitude homology and path homology

    Yasuhiko Asao. Magnitude homology and path homology. Bull. Lond. Math. Soc., 55(1):375– 398, 2023

    2023

  4. [4]

    Homotopy theory of graphs

    Eric Babson, H ´el`ene Barcelo, Mark de Longueville, and Reinhard Laubenbacher. Homotopy theory of graphs. J. Algebraic Combin., 24(1):31–44, 2006

    2006

  5. [5]

    Kenneth S. Brown. Abstract homotopy theory and generalized sheaf cohomology. Trans. Amer. Math. Soc., 186:419–458, 1973

    1973

  6. [6]

    Cubical setting for discrete homotopy theory, revis- ited

    Daneil Carranza and Krzysztof Kapulkin. Cubical setting for discrete homotopy theory, revis- ited. Compositio Mathematica, 160(12):2856–2903, 2024

    2024

  7. [7]

    Cofibration category of digraphs for path homology.Algebr

    Daniel Carranza, Brandon Doherty, Krzyzstof Kapulkin, Morgan Opie, Maru Sarazola, and Liang Ze Wong. Cofibration category of digraphs for path homology.Algebr. Comb., 7(2):475– 514, 2024

    2024

  8. [8]

    Nonexistence of colimits in naive dis- crete homotopy theory

    Daniel Carranza, Krzysztof Kapulkin, and Jinho Kim. Nonexistence of colimits in naive dis- crete homotopy theory. Appl. Categ. Struct., 31(5), 2023. Article number 41

    2023

  9. [9]

    Homological Algebra

    Henri Cartan and Samuel Eilenberg. Homological Algebra. Princeton University Press, Prince- ton, NJ, 1956

    1956

  10. [10]

    Derived A- infinity algebras and their homotopies

    Joana Cirici, Daniela Egas Santander, Muriel Livernet, and Sarah Whitehouse. Derived A- infinity algebras and their homotopies. Topology Appl., 235:214–268, 2018

    2018

  11. [11]

    Model category structures and spectral sequences

    Joana Cirici, Daniela Egas Santander, Muriel Livernet, and Sarah Whitehouse. Model category structures and spectral sequences. Proc. Roy. Soc. Edinburgh Sect. A, 150(6):2815–2848, 2020. HOMOTOPY THEORIES VIA THE MPSS 34

    2020

  12. [12]

    Cat ´egories d´erivables

    Denis-Charles Cisinski. Cat ´egories d´erivables. Bull. Soc. Math. France, 138(3):317–393, 2010

    2010

  13. [13]

    Th ´eorie de Hodge

    Pierre Deligne. Th ´eorie de Hodge. II. Inst. Hautes ´Etudes Sci. Publ. Math., 40:5–57, 1971

    1971

  14. [14]

    Ivanov, Lev Mukoseev, and Mengmeng Zhang

    Shaobo Di, Sergei O. Ivanov, Lev Mukoseev, and Mengmeng Zhang. On the path homology of Cayley digraphs and covering digraphs. J. Algebra, 653:156–199, 2024

    2024

  15. [15]

    The discrete homotopy hypothesis for directed graphs

    Briony Eldridge, Sergei O. Ivanov, Xiomeng Xu, Shing-Tung Yau, and Mengmeng Zhang. The discrete homotopy hypothesis for directed graphs. arXiv:2605.04959, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  16. [16]

    Model category structures on multi- complexes

    Xin Fu, Ai Guan, Muriel Livernet, and Sarah Whitehouse. Model category structures on multi- complexes. Topology Appl., 316:Paper No. 108104, 26, 2022

    2022

  17. [17]

    Algebraic & Geometric Topology , year =

    Kiyonori Gomi. Magnitude homology of geodesic space. arXiv:1902.07044, 2019

    work page arXiv 1902

  18. [18]

    On the path homology theory of digraphs and Eilenberg-Steenrod axioms

    Alexander Grigor’yan, Rolando Jimenez, Yuri Muranov, and Shing-Tung Yau. On the path homology theory of digraphs and Eilenberg-Steenrod axioms. Homology Homotopy Appl. , 20(2):179–205, 2018

    2018

  19. [19]

    Homotopy theory for digraphs

    Alexander Grigor’yan, Yong Lin, Yuri Muranov, and Shing-Tung Yau. Homotopy theory for digraphs. Pure Appl. Math. Q., 10(4):619–674, 2014

    2014

  20. [20]

    Homologies of path complexes and digraphs

    Alexander Grigor’yan, Yong Lin, Yuri Muranov, and Shing-Tung Yau. Homologies of path complexes and digraphs. arXiv:1207.2834, 2012

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  21. [21]

    Quantitative homotopy theory

    Mikhael Gromov. Quantitative homotopy theory. In Prospects in mathematics (Princeton, NJ, 1996), pages 45–49. Amer. Math. Soc., Providence, RI, 1999

    1996

  22. [22]

    Algebraic Topology

    Allen Hatcher. Algebraic Topology. Cambridge University Press, Cambridge, 2002

    2002

  23. [23]

    Bigraded path homology and the magnitude-path spectral sequence

    Richard Hepworth and Emily Ro ff. Bigraded path homology and the magnitude-path spectral sequence. arXiv:2404.06689, 2024

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  24. [24]

    The reachability homology of a directed graph

    Richard Hepworth and Emily Ro ff. The reachability homology of a directed graph. Int. Math. Res. Not. IMRN., 2025(3):1–18, 2025. rnae280

    2025

  25. [25]

    Categorifying the magnitude of a graph

    Richard Hepworth and Simon Willerton. Categorifying the magnitude of a graph. Homol. Ho- motopy Appl., 19:31–60, 2017

    2017

  26. [26]

    Sergei O. Ivanov. Nested homotopy models of finite metric spaces and their spectral homology. arXiv:2312.11878, 2023

    work page arXiv 2023

  27. [27]

    The fundamental group in discrete homotopy theory

    Krzysztof Kapulkin and Udit Mavinkurve. The fundamental group in discrete homotopy theory. Advances in Applied Mathematics, 164:102838, 2025

    2025

  28. [28]

    A homology vanishing theorem for graphs with positive curvature

    Mark Kempton, Florentin M ¨unch, and Shing-Tung Yau. A homology vanishing theorem for graphs with positive curvature. Comm. Anal. Geom., 29(6):1449–1473, 2021

    2021

  29. [29]

    The magnitude of a graph.Math

    Tom Leinster. The magnitude of a graph.Math. Proc. Cambridge Philos. Soc., 166(2):247–264, 2019

    2019

  30. [30]

    Magnitude homology of enriched categories and metric spaces

    Tom Leinster and Michael Shulman. Magnitude homology of enriched categories and metric spaces. Algebr. Geom. Topol., 21(5):2175–2221, 2021

    2021

  31. [31]

    Fibration categories are fibrant relative categories

    Lennart Meier. Fibration categories are fibrant relative categories. Algebr. Geom. Topol. , 16(6):3271–3300, 2016

    2016

  32. [32]

    Cofibrations in Homotopy Theory

    Andrei Radulescu-Banu. Cofibrations in homotopy theory. arXiv:math/0610009v4, 2009

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  33. [33]

    Category Theory in Context

    Emily Riehl. Category Theory in Context. Aurora Dover Modern Math Originals. Dover Publi- cations, Inc., Mineola, NY , 2016. ML: Universit´e Paris Cit´e, Sorbonne Universit´e, CNRS, IMJ-PRG, F-75013 Paris, France ER: School of Mathematics and Maxwell Institute for MathematicalSciences, University of Edinburgh, Scotland SW: School of Mathematical andPhysic...

    2016