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arxiv: 2201.06650 · v6 · submitted 2022-01-17 · 🧮 math.AT

Galois Connections in Persistent Homology

Aziz Burak Gulen , Alexander McCleary This is my paper

Pith reviewed 2026-05-24 13:07 UTC · model grok-4.3

classification 🧮 math.AT
keywords persistent homologyGalois connectionsbottleneck stabilityinterleavingsmatchingsmultiparameter persistenceRota's theorem
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The pith

Galois connections unify interleavings and matchings in persistent homology and yield an easier proof of the bottleneck stability theorem via Rota's theorem.

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

The paper develops a language for persistent homology by recasting its central constructions as Galois connections between posets. This recasting unifies interleavings and matchings into a single framework and grants direct access to Rota's Galois connection theorem. Using that theorem, the authors derive the bottleneck stability theorem without the usual technical overhead. A sympathetic reader would care because the same language also relates distinct notions of multiparameter persistence diagrams, suggesting that many stability and comparison results may admit similar simplifications.

Core claim

By expressing the standard notions of interleavings and matchings as Galois connections between appropriate posets of persistence modules and diagrams, the constructions preserve the categorical and metric structures needed for Rota's theorem to apply directly, thereby supplying a substantially shorter proof of the bottleneck stability theorem while also establishing explicit relationships among various multiparameter persistence diagrams.

What carries the argument

Galois connections between posets of persistence modules and diagrams that preserve the interleaving and matching structures.

If this is right

  • The bottleneck stability theorem follows immediately once the relevant Galois connections are identified.
  • Different notions of multiparameter persistence diagrams become comparable through the same poset maps.
  • Any other stability statement that can be phrased in terms of interleavings or matchings becomes eligible for the same reduction.
  • The framework supplies a uniform way to move between module-level and diagram-level descriptions of persistence.

Where Pith is reading between the lines

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

  • The same Galois-connection language may shorten proofs of stability for other distances or for zigzag persistence.
  • Poset-theoretic algorithms for computing Galois connections could be tested directly against existing persistence-diagram software.
  • The approach suggests that order-theoretic invariants might replace some metric arguments in computational topology.

Load-bearing premise

The standard constructions of interleavings and matchings can be realized as Galois connections that preserve the categorical and metric structures required for Rota's theorem.

What would settle it

An explicit pair of persistence modules whose interleaving distance fails to equal the distance induced by the corresponding Galois connection and matching would falsify the modeling step.

Figures

Figures reproduced from arXiv: 2201.06650 by Alexander McCleary, Aziz Burak Gulen.

Figure 1
Figure 1. Figure 1: A filtration indexed by the totally ordered set [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The diagram on the left is an interleaving between two persistence modules [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The Hasse diagram of the poset P from Example 2.2. 2 Preliminaries Here we present some background on partially ordered sets and M¨obius inversion. For a more thorough reference see [17]. 2.1 Partially Ordered Sets Definition 2.1: A partially ordered set, or poset, is a set P with a relation 6 satisfying • Reflexivity: a 6 a for any a ∈ P. • Antisymmetry: if a 6 b and b 6 a then a = b for any a, b ∈ P. • T… view at source ↗
Figure 4
Figure 4. Figure 4: A persistence module over P; see Example 4.7 Remark 4.4: Note that there are no constructible functors defined on R. This is due to the fact that there are no co-closure operators with finite images on R. There are, however, plenty of constructible functors defined on the posets [−∞,∞], [−∞,∞) [0, ∞], and [0, ∞). We now define a category of persistence modules. The objects are functors from posets to the c… view at source ↗
Figure 5
Figure 5. Figure 5: A free presentation of the persistence module [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: An interleaving Γ between the two persistence modules M0 and M1. We interpret Γ as a persistence module with marginals M0 and M1. See Example 6.3 for more details. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: This example shows that nonnegativity is crucial in the definition of a matching. See [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Above are three persistence diagrams w1, w2, and w3 with matchings ν between w1 and w2, and η between w2 and w3. There is no unique way to glue the matchings ν and η together. Instead, our algorithm proceeds by choosing (1, 3) in the support of w2 and choosing points (say, (0, 3) and (2, 3)) matched to (1, 3) by ν and η respectively. The algorithm then matches (0, 3) with (2, 3) and removes them from the p… view at source ↗
Figure 9
Figure 9. Figure 9: The interpolation of M0 and M1 induced by the interleaving Γ from Example 6.3. The domain St of Mt is given by the intersection of the dashed lines with the vertical line at t. The critical points of the interpolation occur where the dashed lines intersect: at 0, .5, and 1. The maps αt,c flow along the dashed lines. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The inclusion ι of the line ` into [0, ∞) 2 and its left adjoint π. Proposition 8.4: The persistence diagram of the 1-parameter persistence module M|` is given by [π](∂m)]. Proof. We have that M|` =∼ M◦ ι which is a morphism from M to M` and so, by functoriality, [π](∂m)] = [∂(m|`)]. In particular, this proof gives a linear time algorithm for computing the fibered barcode from the birth-death function per… view at source ↗
read the original abstract

We present a new language for persistent homology in terms of Galois connections. This language has two main advantages over traditional approaches. First, it simplifies and unifies central concepts such as interleavings and matchings. Second, it provides access to Rota's Galois connection theorem -- a powerful tool with many potential applications in applied topology. To illustrate this, we use Rota's Galois connection theorem to give a substantially easier proof of the bottleneck stability theorem. Finally, we use this language to establish relationships between various notions of multiparameter persistence diagrams.

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 / 3 minor

Summary. The paper introduces a language for persistent homology based on Galois connections between posets. This is claimed to simplify and unify interleavings and matchings, to grant access to Rota's Galois connection theorem, and thereby to yield a substantially easier proof of the bottleneck stability theorem; the same language is also used to relate different notions of multiparameter persistence diagrams.

Significance. If the poset constructions preserve the required categorical and metric structures, the work supplies a unifying order-theoretic perspective that simplifies central definitions and imports a classical theorem for a shorter stability proof. The explicit use of an external result (Rota's theorem) once the Galois connections are verified is a methodological strength that could extend to other questions in applied topology.

minor comments (3)
  1. [Abstract] The abstract asserts a 'substantially easier proof' but does not indicate the section or theorem number where the new argument appears; a forward reference would help readers locate the comparison.
  2. Notation for the various posets and their Galois connections is introduced incrementally; a single summary table or diagram collecting the objects, maps, and induced distances would improve readability.
  3. A few sentences in the multiparameter section use the same symbol for distinct but related diagrams; consistent subscripting or a short glossary would remove ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper recasts standard interleavings and matchings in persistent homology as Galois connections between posets, then invokes the external classical result Rota's Galois connection theorem to obtain the bottleneck stability theorem. This structure does not reduce the target theorem to any quantity defined inside the paper by construction, nor does it rely on self-citation chains, fitted parameters renamed as predictions, or ansatzes smuggled via prior work by the same authors. The central move is an external theorem application whose validity hinges on the (independent) verification that the poset maps are indeed Galois connections preserving the relevant metric data; that verification step is presented as the novel contribution rather than presupposed. No load-bearing step collapses to a self-referential definition or renaming of a known result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the existence and standard properties of Galois connections and on Rota's theorem from order theory; no free parameters or invented entities are indicated in the abstract.

axioms (1)
  • standard math Rota's Galois connection theorem holds in the poset setting used for persistence modules.
    Invoked to obtain the easier proof of bottleneck stability.

pith-pipeline@v0.9.0 · 5607 in / 1126 out tokens · 22032 ms · 2026-05-24T13:07:51.874452+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Interleaving Distance as a Galois-Edit Distance

    math.AT 2025-09 unverdicted novelty 7.0

    Interleaving distance on single- and multi-parameter persistence modules equals a Galois-edit distance, yielding a new proof of bottleneck stability.

Reference graph

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