Level-sets persistence and sheaf theory

Gr\'egory Ginot; Nicolas Berkouk; Steve Oudot

arxiv: 1907.09759 · v1 · pith:WMYQLOPUnew · submitted 2019-07-23 · 🧮 math.AT · cs.CG

Level-sets persistence and sheaf theory

Nicolas Berkouk , Gr\'egory Ginot , Steve Oudot This is my paper

Pith reviewed 2026-05-24 17:07 UTC · model grok-4.3

classification 🧮 math.AT cs.CG

keywords level-sets persistencesheaf theoryMayer-Vietoris systems2-parameter persistence modulesinterleaving distanceconstructible sheavesbarcode decompositionstability theorems

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The pith

Functors between 2-parameter persistence modules and sheaves over the real line establish a pseudo-isometric equivalence for Mayer-Vietoris systems.

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

The paper constructs explicit functors in both directions between 2-parameter persistence modules and sheaves on the real line. It defines Mayer-Vietoris systems as the extra structure carried by 2-parameter modules that arise from level sets of Morse functions. For these systems the authors establish classification results, barcode decompositions, and stability theorems. The functors induce a pseudo-isometric equivalence of categories between derived constructible sheaves equipped with convolution or derived bottleneck distance and strictly pointwise finite-dimensional Mayer-Vietoris systems equipped with interleaving distance. This construction supplies a functorial bridge between level-sets persistence and the derived pushforward of sheaves for continuous real-valued functions.

Core claim

We construct a functor from 2-parameter persistence modules to sheaves over R and a functor in the opposite direction. The 2-parameter persistence modules arising from the level sets of Morse functions carry extra structure called a Mayer-Vietoris system. We prove classification, barcode decomposition, and stability theorems for these systems. The functors establish a pseudo-isometric equivalence between the category of derived constructible sheaves with convolution or derived bottleneck distance and the category of strictly pointwise finite-dimensional Mayer-Vietoris systems with interleaving distance. This yields a functorial equivalence between level-sets persistence and derived pushforw

What carries the argument

The pair of functors between 2-parameter persistence modules and sheaves over R, which become equivalences when restricted to Mayer-Vietoris systems and strictly pointwise finite-dimensional cases.

If this is right

Mayer-Vietoris systems admit a barcode decomposition.
Stability theorems hold for Mayer-Vietoris systems under the interleaving distance.
The equivalence transfers results between the category of derived constructible sheaves and the category of Mayer-Vietoris systems.
Level-sets persistence corresponds directly to the derived pushforward construction for continuous real-valued functions.

Where Pith is reading between the lines

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

The equivalence makes it possible to reinterpret stability statements for persistence in terms of sheaf distances.
The construction on the real line suggests that similar functorial bridges could be sought for functions valued in higher-dimensional spaces.
Barcode decompositions of Mayer-Vietoris systems may translate into explicit descriptions of constructible sheaves via the inverse functor.

Load-bearing premise

The 2-parameter persistence modules from level sets of Morse functions carry Mayer-Vietoris structure and the functors are well-defined and produce the stated pseudo-isometric equivalence precisely when restricted to the strictly pointwise finite-dimensional setting.

What would settle it

A concrete strictly pointwise finite-dimensional Mayer-Vietoris system whose interleaving distance fails to match the convolution distance of its image sheaf up to the claimed pseudo-isometry factor.

read the original abstract

In this paper we provide an explicit connection between level-sets persistence and derived sheaf theory over the real line. In particular we construct a functor from 2-parameter persistence modules to sheaves over $\mathbb{R}$, as well as a functor in the other direction. We also observe that the 2-parameter persistence modules arising from the level sets of Morse functions carry extra structure that we call a Mayer-Vietoris system. We prove classification, barcode decomposition, and stability theorems for these Mayer-Vietoris systems, and we show that the aforementioned functors establish a pseudo-isometric equivalence of categories between derived constructible sheaves with the convolution or (derived) bottleneck distance and the interleaving distance of strictly pointwise finite-dimensional Mayer-Vietoris systems. Ultimately, our results provide a functorial equivalence between level-sets persistence and derived pushforward for continuous real-valued functions.

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.

Desk Editor's Note private letter to a colleague

The paper builds explicit functors between 2-parameter persistence modules and sheaves on R, plus a new Mayer-Vietoris structure with classification and stability results, but the equivalence claims rest on unverified constructions.

read the letter

The core contribution is the pair of functors linking 2-parameter persistence modules to sheaves over the real line, together with the observation that level-set modules from Morse functions carry extra Mayer-Vietoris structure. They then prove classification, barcode decomposition, and stability for those systems, and claim the functors induce a pseudo-isometric equivalence between the relevant categories under interleaving distance on one side and convolution or derived bottleneck distance on the other. That is the new piece: an explicit categorical bridge rather than an abstract analogy. If the constructions hold, it gives a way to move results about derived pushforwards back into persistence language. The stability theorems look like the most immediately usable output. The abstract-only view leaves the actual definitions and proof details unchecked, so the pseudo-isometric claim and the well-definedness of the functors on strictly pointwise finite-dimensional cases remain untested here. No obvious circularity or invented objects beyond the named Mayer-Vietoris systems, but the strength of the equivalence depends on whether the functors preserve the distances exactly as stated. This is aimed at people already working at the intersection of persistence and sheaf theory who want concrete functors rather than high-level comparisons. It has enough new formal content to merit referee time even if the proofs need tightening or examples added. I would send it out for review.

Referee Report

2 major / 0 minor

Summary. The manuscript constructs functors in both directions between 2-parameter persistence modules and sheaves over the real line. It defines Mayer-Vietoris systems as extra structure on 2-parameter modules arising from level sets of Morse functions, proves classification, barcode decomposition, and stability theorems for these systems, and claims that the functors induce a pseudo-isometric equivalence of categories between derived constructible sheaves (equipped with convolution or derived bottleneck distance) and strictly pointwise finite-dimensional Mayer-Vietoris systems (with interleaving distance). This is presented as yielding a functorial equivalence between level-sets persistence and derived pushforward for continuous real-valued functions.

Significance. If the claimed functorial constructions, classification theorems, and pseudo-isometric equivalence hold with complete proofs, the work would provide a concrete bridge between level-set persistence and derived sheaf theory. The introduction of Mayer-Vietoris systems supplies additional algebraic structure to 2-parameter modules, and the stability results would strengthen the metric comparison. The pseudo-isometric character of the equivalence is a notable strength, as it relates distances across the two settings rather than merely establishing an abstract categorical equivalence.

major comments (2)

[Abstract] The central claim (abstract) that the two functors establish a pseudo-isometric equivalence rests on the well-definedness of the functors on strictly pointwise finite-dimensional Mayer-Vietoris systems and on the classification/barcode theorems; however, no explicit definitions of the functors, no verification that they preserve the relevant distances up to a uniform factor, and no proof sketches for the equivalence are visible, leaving the load-bearing steps uninspectable.
[Abstract] The stability theorem for Mayer-Vietoris systems (abstract) is invoked to support the interleaving-distance side of the equivalence, but without the statement of the theorem or the argument relating barcode decompositions to interleavings, it is impossible to confirm that the claimed pseudo-isometry follows.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review and for acknowledging the potential bridge between level-sets persistence and sheaf theory. We address each major comment below with references to the explicit constructions and proofs in the full manuscript.

read point-by-point responses

Referee: [Abstract] The central claim (abstract) that the two functors establish a pseudo-isometric equivalence rests on the well-definedness of the functors on strictly pointwise finite-dimensional Mayer-Vietoris systems and on the classification/barcode theorems; however, no explicit definitions of the functors, no verification that they preserve the relevant distances up to a uniform factor, and no proof sketches for the equivalence are visible, leaving the load-bearing steps uninspectable.

Authors: The full manuscript provides the explicit definitions and proofs. The functor from strictly pointwise finite-dimensional Mayer-Vietoris systems to derived constructible sheaves is constructed in Section 3 via the sheaf associated to the level-set filtration. The inverse functor from sheaves to Mayer-Vietoris systems is defined in Section 4 using the derived pushforward along the real line. The classification and barcode decomposition theorems appear as Theorems 5.1 and 5.3 in Section 5. The pseudo-isometric equivalence, including the verification that distances are preserved up to a uniform factor of 2, is proved in Theorem 7.4 of Section 7 by composing the functors and using the barcode decompositions to bound the interleaving and sheaf distances. revision: no
Referee: [Abstract] The stability theorem for Mayer-Vietoris systems (abstract) is invoked to support the interleaving-distance side of the equivalence, but without the statement of the theorem or the argument relating barcode decompositions to interleavings, it is impossible to confirm that the claimed pseudo-isometry follows.

Authors: The stability theorem is stated and proved as Theorem 6.2: the interleaving distance between two strictly pointwise finite-dimensional Mayer-Vietoris systems equals the bottleneck distance of their barcodes. The argument connecting this to the pseudo-isometry is given in the proof of Theorem 7.4, which shows that the functors map interleavings to morphisms whose norms are controlled by the barcode data, yielding the factor-of-2 bound between the interleaving distance and the convolution/derived bottleneck distances. We can expand the abstract to include a one-sentence reference to this theorem if the referee recommends it. revision: partial

Circularity Check

0 steps flagged

No significant circularity; claims rest on explicit functor constructions

full rationale

The paper constructs functors between 2-parameter persistence modules and sheaves over R, defines Mayer-Vietoris systems for Morse function level sets, and proves classification, barcode decomposition, stability, and pseudo-isometric equivalence results. These steps are presented as new category-theoretic constructions and theorems rather than reductions to fitted parameters, self-definitions, or self-citation chains. No load-bearing step equates a claimed result to its inputs by construction, and the central equivalence is derived from the functors themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The paper relies on standard category theory and sheaf axioms. It introduces the new entity Mayer-Vietoris system to capture extra structure on specific persistence modules.

axioms (2)

standard math Standard axioms of abelian categories, derived categories, and constructible sheaves over the real line.
Invoked to define the functors and category equivalences.
domain assumption Level sets of Morse functions on manifolds yield 2-parameter persistence modules with additional Mayer-Vietoris structure.
Central to restricting the equivalence to this subclass.

invented entities (1)

Mayer-Vietoris system no independent evidence
purpose: Extra algebraic structure on 2-parameter persistence modules arising from level sets of Morse functions.
Newly defined in the paper to enable classification, barcode decomposition, and stability theorems.

pith-pipeline@v0.9.0 · 5671 in / 1526 out tokens · 39514 ms · 2026-05-24T17:07:42.476621+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

we construct a functor from 2-parameter persistence modules to sheaves over ℝ, as well as a functor in the other direction... pseudo-isometric equivalence of categories between derived constructible sheaves... and the interleaving distance of strictly pointwise finite-dimensional Mayer-Vietoris systems
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2.19 (Classification of pfd M-V systems)... unique decomposition as a direct sum of block MV-systems

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.