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arxiv: 2606.23859 · v1 · pith:DWRTFZPTnew · submitted 2026-06-22 · 💻 cs.CG

Canopies: A Generalization of Vines and Vineyards for Parameterized Persistence

Barbara Giunti , Elizabeth Munch This is my paper

Pith reviewed 2026-06-26 05:39 UTC · model grok-4.3

classification 💻 cs.CG
keywords canopiesparameterized persistencevinesvineyardspersistent homologyfiltered chain complexesmonodromy
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The pith

Canopies track choices of simplex pairs in filtered complexes rather than persistence diagram points, producing homeomorphic structures for any filtration.

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

The paper introduces canopies as a construction for parameterized persistence obtained by replacing each point in the persistence diagram with a choice of the simplices that created it. Two versions exist: the A-canopy keeps information about diagonal points while the D-canopy matches the usual diagram. The algebraic structure of filtered chain complexes is used to prove that any two valid choices of pairs yield homeomorphic canopies. This viewpoint supports defining vines when points have multiplicity, discussing monodromy, and relating non-trivial monodromy in the persistent homology transform to non-Hausdorff points in the canopy.

Core claim

By tracking a choice of simplex pairs instead of the output point in the persistence diagram, a canopy can be built directly from any filtered filtration function. The algebraic structure of filtered chain complexes guarantees that different valid pair choices produce homeomorphic canopies. This combinatorial replacement of the persistence bundle enables vines with multiplicity and a direct connection between non-trivial monodromy in the persistent homology transform and non-Hausdorff points in the canopy.

What carries the argument

The canopy, obtained by replacing each persistence point with a choice of the pair of simplices whose birth and death generate it, acting as a combinatorial encoding of the filtered chain complex.

If this is right

  • Vines can be defined even when persistence points occur with multiplicity.
  • Monodromy of the parameterized persistence can be discussed directly from the canopy.
  • Non-trivial monodromy in the persistent homology transform implies the existence of non-Hausdorff points in the canopy.

Where Pith is reading between the lines

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

  • The same pair-tracking idea could be used to extend other persistence summaries that currently discard diagonal or multiplicity information.
  • Because the canopy is built from the chain complex level, it may support direct comparison of persistence across different filtrations without first computing diagrams.

Load-bearing premise

The algebraic structure of filtered chain complexes is enough to force homeomorphism between canopies built from different valid simplex pair choices.

What would settle it

An explicit filtered complex together with two different valid simplex pair selections that produce non-homeomorphic canopies would falsify the homeomorphism claim.

Figures

Figures reproduced from arXiv: 2606.23859 by Barbara Giunti, Elizabeth Munch.

Figure 1
Figure 1. Figure 1: Example adapted from [3]. At left is the simplicial complex L used as a running example. At right are three functions f, g, h: K → R specified on edges a and b, and triangles c and d. Below each function is the Hasse diagram of the minimal compatible orderings ≺f , ≺g, ≺h for the named simplices. We assume that all other vertices and edges have function value 0 but generally ignore them for the sake of our… view at source ↗
Figure 2
Figure 2. Figure 2: Left is the running example simplicial complex [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: An cartoon example of an A-canopy (see Definition 3.1). Remark 3.3. In a sense, we are working in the following diagram: Fibered Filt. Functions K × B → R Func(B, DGMA) A-canopies induced Thm. 3.8 induced Obtaining the induced B-parametrized diagram from an A-canopy is straightforward from the set-up, hence the use of the term the induced B-parameterized diagram. Moreover, it also follows immediately that … view at source ↗
Figure 4
Figure 4. Figure 4: A sequence of functions on a simplicial complex [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A fibered filtration function for the simplicial complex [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Left: The space E for the A-canopy of the fibered filtration function given in [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: A cartoon of a D-canopy (Definition 4.1), modified from [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The D-canopy (right) for an input fibered filtration function given by the PHT of the embedded [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Left is the running example simplicial complex [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
read the original abstract

In this paper, we provide a new construction for studying parameterized persistence, called a canopy. We give two versions of this construction: the A-canopy, retaining all information about points on the diagonal of the persistence diagram; and the D-canopy, encoding the information of the "standard" persistence diagram. We do this by making a simple but major modification in the persistence bundle representation information: namely, rather than tracking a point in the persistence diagram, we instead track some choice of pairs of simplices that created said point. This viewpoint is a combinatorial version of tracking the chain complex information rather than just the output of persistence. We show how to construct the canopies from any filtered filtration function, proving, using the algebraic structure of filtered chain complexes, that different choices of pairs result in homeomorphic structures. Finally, we showcase the power of our approach by using canopies to define vines even in the presence of points with multiplicity; to discuss monodromy; and to obtain some immediate results linking non-trivial monodromy in the persistent homology transform with the existence of non-Hausdorff points in the canopy.

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

1 major / 2 minor

Summary. The manuscript introduces canopies as a generalization of vines and vineyards for studying parameterized persistence. It defines two versions: the A-canopy, which retains information about diagonal points, and the D-canopy, which encodes standard persistence diagram information. By tracking choices of simplex pairs instead of persistence diagram points, the authors construct canopies from any filtered filtration function. They prove that different pair choices yield homeomorphic structures using the algebraic properties of filtered chain complexes. Applications include defining vines with multiplicity, analyzing monodromy, and relating non-trivial monodromy in the persistent homology transform to non-Hausdorff points in the canopy.

Significance. If the central homeomorphism result holds with the required structure-preserving properties, this work provides a valuable combinatorial framework for parameterized persistence that extends existing concepts like vines to handle multiplicities and monodromy. The use of algebraic structure of filtered chain complexes for the proof is a positive aspect, offering a more foundational approach than purely combinatorial methods.

major comments (1)
  1. [Homeomorphism proof (algebraic argument using filtered chain complexes)] The homeomorphism claim (abstract and the section presenting the algebraic proof via filtered chain complexes) establishes equivalence of total spaces for different simplex-pair choices, but does not explicitly establish that the homeomorphism is fiberwise over the filtration parameter or that it induces a correspondence on the tracked pairs at each parameter value. This is load-bearing for the applications to vines with multiplicity and monodromy, as a non-fiberwise homeomorphism could fail to preserve the combinatorial data needed for those claims.
minor comments (2)
  1. [Construction section] Notation for A-canopy versus D-canopy could be introduced with a short table or diagram in the construction section to clarify the distinction between retaining diagonal information and encoding the standard diagram.
  2. [Abstract] The abstract states the homeomorphism result but does not preview whether it is fiberwise; adding one sentence would better connect the proof to the later applications.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting an important point about the homeomorphism. We address the major comment below.

read point-by-point responses

  1. Referee: The homeomorphism claim (abstract and the section presenting the algebraic proof via filtered chain complexes) establishes equivalence of total spaces for different simplex-pair choices, but does not explicitly establish that the homeomorphism is fiberwise over the filtration parameter or that it induces a correspondence on the tracked pairs at each parameter value. This is load-bearing for the applications to vines with multiplicity and monodromy, as a non-fiberwise homeomorphism could fail to preserve the combinatorial data needed for those claims.

    Authors: We agree that the fiberwise property over the filtration parameter, together with the induced correspondence on tracked pairs, is essential for the applications to multiplicity and monodromy. The algebraic argument in the manuscript proceeds from the structure of filtered chain complexes, whose chain groups, boundary maps, and induced isomorphisms are all defined levelwise with respect to the filtration parameter. Consequently the homeomorphism between total spaces is fiberwise and preserves the pairing data at each parameter value. To make this explicit and to strengthen the link to the applications, we will add a clarifying statement (and, if space permits, a short lemma) in the section containing the algebraic proof. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on standard algebraic properties of filtered chain complexes

full rationale

The paper constructs canopies by tracking simplex pairs in filtered chain complexes and proves homeomorphism of structures for different pair choices using the algebraic structure of those complexes. This is presented as an application of existing algebraic topology facts rather than a self-referential definition, fitted parameter renamed as prediction, or load-bearing self-citation. No equations or steps reduce the claimed homeomorphism or applications (vines with multiplicity, monodromy) to the inputs by construction. The central result is self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are stated in the provided text.

pith-pipeline@v0.9.1-grok · 5722 in / 1073 out tokens · 29273 ms · 2026-06-26T05:39:12.660131+00:00 · methodology

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