A sheaf-theoretic approach to tropical homology

Andreas Gross; Farbod Shokrieh

arxiv: 1906.09245 · v1 · pith:GEXENENDnew · submitted 2019-06-21 · 🧮 math.AG · math.CO

A sheaf-theoretic approach to tropical homology

Andreas Gross , Farbod Shokrieh This is my paper

Pith reviewed 2026-05-25 18:27 UTC · model grok-4.3

classification 🧮 math.AG math.CO MSC 14T05

keywords tropical homologysheaf theoryBorel-Moore homologyPoincaré-Verdier dualitytropical manifoldscycle class mapKünneth theoremtropical geometry

0 comments

The pith

A sheaf-theoretic definition equips tropical homology with proper push-forwards, products, and Poincaré-Verdier duality over the integers.

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

The paper defines tropical homology via sheaves to accommodate non-compact supports. This construction is shown to support the same formal operations as classical Borel-Moore homology, including proper push-forwards, cross products, cup products, the projection formula, and the Künneth theorem. The authors also introduce a tropical cycle class map that is a natural transformation and prove Poincaré-Verdier duality over the integers for tropical manifolds. A reader would care because the setup turns tropical homology into a functorial theory that can interact with maps and products in a controlled way.

Core claim

The sheaf-theoretic approach to tropical homology reproduces the expected groups on polyhedral complexes and behaves analogously to classical Borel-Moore homology by admitting proper push-forwards, cross products, cup products with tropical cohomology classes, the projection formula, and the Künneth theorem. The framework supplies a natural definition of the tropical cycle class map as a natural transformation, and Poincaré-Verdier duality holds over the integers on tropical manifolds.

What carries the argument

The sheaf-theoretic definition of tropical homology, which extends the groups to non-compact supports and carries the functorial operations and duality.

If this is right

Proper push-forwards exist along proper maps of tropical spaces.
Cross products and cup products with cohomology classes are defined and obey the projection formula.
The Künneth theorem holds for the tropical homology groups.
The tropical cycle class map is a natural transformation between appropriate functors.
Poincaré-Verdier duality holds over the integers for tropical manifolds.

Where Pith is reading between the lines

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

The sheaf construction could allow spectral sequences or local-to-global arguments to compute tropical homology in new cases.
The parallel with Borel-Moore homology suggests that tropical geometry can now import standard topological tools for handling non-compact spaces.
Integral duality may let tropical intersection theory work with integer coefficients without additional torsion issues.
Functoriality could support the study of morphisms and families of tropical varieties in a uniform categorical setting.

Load-bearing premise

The sheaf-theoretic definition reproduces the standard tropical homology groups on polyhedral complexes and satisfies the axioms required for the functorial properties and duality.

What would settle it

A polyhedral complex where the groups computed from the sheaf definition differ from the known combinatorial tropical homology groups, or a tropical manifold where Poincaré-Verdier duality fails over the integers.

read the original abstract

We introduce a sheaf-theoretic approach to tropical homology, especially for tropical homology with potentially non-compact supports. Our setup is suited to study the functorial properties of tropical homology, and we show that it behaves analogously to classical Borel-Moore homology in the sense that there are proper push-forwards, cross products, and cup products with tropical cohomology classes, and that it satisfies identities like the projection formula and the K\"unneth theorem. Our framework allows for a natural definition of the tropical cycle class map, which we show to be a natural transformation. Finally, we prove Poincar\'e-Verdier duality over the integers on tropical manifolds.

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 gives a sheaf-theoretic definition of tropical homology with non-compact supports and proves the expected functoriality plus integral Poincaré-Verdier duality.

read the letter

The main takeaway is that Gross and Shokrieh define tropical homology via sheaves in a way that handles non-compact supports and then verify it satisfies the usual properties of Borel-Moore homology: proper push-forwards, cross products, cup products with cohomology classes, the projection formula, the Künneth theorem, a natural cycle class map, and Poincaré-Verdier duality over the integers on tropical manifolds. This is the part that stands out relative to earlier combinatorial treatments of tropical homology. The framework organizes these identities in one place and makes the functoriality look more systematic. The cycle class map being natural is a clean addition. The paper does the basic work of checking that the new definition recovers the expected groups on polyhedral complexes, which is the modeling step everything else rests on. Once that holds, the rest follows from standard sheaf-theoretic arguments rather than ad-hoc combinatorial proofs. A minor soft spot is that the abstract does not spell out how much new computation is needed to establish the duality over Z versus how much is inherited from the classical Verdier duality once the sheaf is set up correctly. If the verification of the definition is careful, this is not a problem, but it is the place where a referee would look first. The paper is aimed at people already working in tropical geometry who need these homological tools for further applications. It is a focused technical contribution rather than a broad reworking of the field. I would send it to peer review; the claims are concrete enough that referees can check the definition and the listed identities without needing to guess at the authors' intent.

Referee Report

0 major / 2 minor

Summary. The paper introduces a sheaf-theoretic definition of tropical homology (with emphasis on non-compact supports) and shows that the resulting theory admits proper push-forwards, cross products, cup products with tropical cohomology classes, the projection formula, and the Künneth theorem. It further constructs a tropical cycle class map that is a natural transformation and proves Poincaré-Verdier duality over the integers for tropical manifolds.

Significance. If the sheaf-theoretic construction reproduces the expected groups on polyhedral complexes and satisfies the listed axioms, the framework supplies a categorical setting in which functoriality of tropical homology can be studied uniformly, mirroring the classical theory of Borel-Moore homology. The integral duality statement is a concrete strengthening of existing comparisons between tropical and algebraic geometry.

minor comments (2)

[Introduction / §2] The abstract states that the new definition reproduces the expected groups on polyhedral complexes, but the introduction or §2 should contain an explicit comparison (e.g., a proposition or remark) verifying agreement with the combinatorial definition of tropical homology on a polyhedral complex; this verification is load-bearing for all subsequent functoriality claims.
[Abstract] Notation for the sheaf of tropical chains (or the coefficient sheaf) is introduced without a displayed definition in the abstract; a short displayed formula or reference to the precise site on which the sheaf is defined would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The report accurately summarizes the main contributions: the sheaf-theoretic definition of tropical homology (with non-compact supports), the establishment of proper pushforwards, cross and cup products, the projection formula, the Künneth theorem, the natural cycle class map, and Poincaré-Verdier duality over the integers on tropical manifolds.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces a sheaf-theoretic definition of tropical homology (with non-compact supports) and derives its functorial properties (proper push-forwards, cross products, cup products, projection formula, Künneth theorem) plus Poincaré-Verdier duality directly from this modeling choice. The derivation chain begins with the new definition and applies standard sheaf-theoretic arguments; no step reduces by construction to fitted parameters, self-citations, or renamed inputs. The abstract and reader's assessment confirm the central claims rest on independent content from the chosen setup rather than circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work is a new definitional framework in algebraic geometry; it relies on standard sheaf axioms and the definition of tropical manifolds but introduces no fitted numerical parameters or new postulated entities.

axioms (2)

standard math Standard axioms of sheaves on topological spaces and the definition of Borel-Moore homology in the classical setting.
Invoked when the paper states that the new construction behaves analogously to classical Borel-Moore homology.
domain assumption Tropical manifolds are locally modeled on polyhedral complexes with the usual fan structure.
Required for the duality statement to make sense.

pith-pipeline@v0.9.0 · 5625 in / 1395 out tokens · 20675 ms · 2026-05-25T18:27:12.647614+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 1 internal anchor

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[AK06] Federico Ardila and Caroline J. Klivans, The Bergman complex of a matroid and phylogenetic trees, J. Combin. Theory Ser. B 96 (2006), no. 1, 38–49. MR2185977↑39 [AR10] Lars Allermann and Johannes Rau, First steps in tropical intersection theory, Math. Z. 264 (2010), no. 3, 633–670. MR2591823↑3, 9, 10, 12 [Be08] A. Borel et al., Intersection cohomol...

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MR1322960↑22 [FR13] Georges François and Johannes Rau, The diagonal of tropical matroid varieties and cycle intersections, Collect

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Available at http://faculty.tcu.edu/gfriedman/IHbook.pdf.↑49 56 ANDREAS GROSS AND FARBOD SHOKRIEH [GKM09] Andreas Gathmann, Michael Kerber, and Hannah Markwig, Tropical fans and the moduli spaces of tropical curves, Compos. Math. 145 (2009), no. 1, 173–195. MR2480499↑11 [Har67] Robin Hartshorne, Local cohomology, A seminar given by A. Grothendieck, Harvar...

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MR0224620↑54 [IKMZ19] Ilia Itenberg, Ludmil Katzarkov, Grigory Mikhalkin, and Ilia Zharkov, Tropical Homology, Math. Ann. 374 (2019), no. 1-2, 963–1006. MR3961331↑1, 3, 27 [Ive86] Birger Iversen, Cohomology of sheaves , Universitext, Springer-Verlag, Berlin,

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MR842190↑15, 20, 21, 23, 25, 26 [JRS18] Philipp Jell, Johannes Rau, and Kristin Shaw, Lefschetz (1,1)-theorem in tropical geometry, Épijournal Geom. Algébrique 2 (2018), Art. 11,

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MR3894860↑1, 2, 3, 4, 14, 27, 36, 38, 40 [JSS19] Philipp Jell, Kristin Shaw, and Jascha Smacka, Superforms, tropical cohomology, and Poincaré duality, Adv. Geom. 19 (2019), no. 1, 101–130. MR3903579↑3 [KS94] Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds, Grundlehren der Mathematis- chen Wissenschaften, vol. 292, Springer-Verlag, Berlin,

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MR3287221 ↑39 [MZ08] Grigory Mikhalkin and Ilia Zharkov, Tropical curves, their Jacobians and theta functions, Curves and abelian varieties, 2008, pp. 203–230. MR2457739↑6, 13 [MZ14] Grigory Mikhalkin and Ilia Zharkov, Tropical eigenwave and intermediate Jacobians, Homo- logical mirror symmetry and tropical geometry, 2014, pp. 309–349. MR3330789↑1, 2, 3, ...

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Discrete Math

Available at https://archive-ouverte.unige.ch/unige:22758/ATTACHMENT01.↑44 [Sha13] Kristin Shaw, A tropical intersection product in matroidal fans, SIAM J. Discrete Math. 27 (2013), no. 1, 459–491. MR3032930↑3, 42, 47 [Sha15] Kristin Shaw, Tropical surfaces,

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Preprint available at arXiv:1506.07407.↑1 [She85] Allen Dudley Shepard, A cellular description of the derived category of a stratiﬁed space, Ph.D. Thesis,

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Swan, The theory of sheaves , Chicago lectures in mathematics, The University of Chicago Press, Chicago and London, 1964.↑16 [Wei94] Charles A

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Gökova Geom

MR1269324↑24 [Zha13] Ilia Zharkov, The Orlik-Solomon algebra and the Bergman fan of a matroid, J. Gökova Geom. Topol. GGT 7 (2013), 25–31. MR3153919↑8 E-mail address: andreas.gross@colostate.edu E-mail address: farbod@math.ku.dk, farbod@uw.edu

work page 2013

[1] [1]

Klivans, The Bergman complex of a matroid and phylogenetic trees, J

[AK06] Federico Ardila and Caroline J. Klivans, The Bergman complex of a matroid and phylogenetic trees, J. Combin. Theory Ser. B 96 (2006), no. 1, 38–49. MR2185977↑39 [AR10] Lars Allermann and Johannes Rau, First steps in tropical intersection theory, Math. Z. 264 (2010), no. 3, 633–670. MR2591823↑3, 9, 10, 12 [Be08] A. Borel et al., Intersection cohomol...

work page 2006

[2] [2]

Notes on the seminar held at the University of Bern, Bern, 1983, Reprint of the 1984 edition. MR2401086↑15, 20 [BIMS15] Erwan Brugallé, Ilia Itenberg, Grigory Mikhalkin, and Kristin Shaw, Brief introduction to tropical geometry, Proceedings of the Gökova Geometry-Topology Conference 2014, 2015, pp. 1–75. MR3381439↑2, 3 [BM60] A. Borel and J. C. Moore, Hom...

work page 1983

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MR1322960↑22 [FR13] Georges François and Johannes Rau, The diagonal of tropical matroid varieties and cycle intersections, Collect

With a view toward algebraic geometry. MR1322960↑22 [FR13] Georges François and Johannes Rau, The diagonal of tropical matroid varieties and cycle intersections, Collect. Math. 64 (2013), no. 2, 185–210. MR3041763↑41 [Fri18] Greg Friedman, Singular intersection homology ,

work page 2013

[4] [4]

Available at http://faculty.tcu.edu/gfriedman/IHbook.pdf.↑49 56 ANDREAS GROSS AND FARBOD SHOKRIEH [GKM09] Andreas Gathmann, Michael Kerber, and Hannah Markwig, Tropical fans and the moduli spaces of tropical curves, Compos. Math. 145 (2009), no. 1, 173–195. MR2480499↑11 [Har67] Robin Hartshorne, Local cohomology, A seminar given by A. Grothendieck, Harvar...

work page 2009

[5] [5]

MR0224620↑54 [IKMZ19] Ilia Itenberg, Ludmil Katzarkov, Grigory Mikhalkin, and Ilia Zharkov, Tropical Homology, Math. Ann. 374 (2019), no. 1-2, 963–1006. MR3961331↑1, 3, 27 [Ive86] Birger Iversen, Cohomology of sheaves , Universitext, Springer-Verlag, Berlin,

work page 2019

[6] [6]

Algébrique 2 (2018), Art

MR842190↑15, 20, 21, 23, 25, 26 [JRS18] Philipp Jell, Johannes Rau, and Kristin Shaw, Lefschetz (1,1)-theorem in tropical geometry, Épijournal Geom. Algébrique 2 (2018), Art. 11,

work page 2018

[7] [7]

MR3894860↑1, 2, 3, 4, 14, 27, 36, 38, 40 [JSS19] Philipp Jell, Kristin Shaw, and Jascha Smacka, Superforms, tropical cohomology, and Poincaré duality, Adv. Geom. 19 (2019), no. 1, 101–130. MR3903579↑3 [KS94] Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds, Grundlehren der Mathematis- chen Wissenschaften, vol. 292, Springer-Verlag, Berlin,

work page 2019

[8] [8]

MR1299726↑15, 43 [Lyu01] V

With a chapter in French by Christian Houzel, Corrected reprint of the 1990 original. MR1299726↑15, 43 [Lyu01] V . V . Lyubashenko,External tensor product of perverse sheaves, Ukraïn. Mat. Zh. 53 (2001), no. 3, 311–322. MR1834663↑41 [Mey11] Henning Meyer, Intersection theory on tropical toric varieties and compactiﬁcations of tropical parameter spaces , P...

work page 1990

[9] [9]

MR3287221 ↑39 [MZ08] Grigory Mikhalkin and Ilia Zharkov, Tropical curves, their Jacobians and theta functions, Curves and abelian varieties, 2008, pp. 203–230. MR2457739↑6, 13 [MZ14] Grigory Mikhalkin and Ilia Zharkov, Tropical eigenwave and intermediate Jacobians, Homo- logical mirror symmetry and tropical geometry, 2014, pp. 309–349. MR3330789↑1, 2, 3, ...

work page 2008

[10] [10]

Discrete Math

Available at https://archive-ouverte.unige.ch/unige:22758/ATTACHMENT01.↑44 [Sha13] Kristin Shaw, A tropical intersection product in matroidal fans, SIAM J. Discrete Math. 27 (2013), no. 1, 459–491. MR3032930↑3, 42, 47 [Sha15] Kristin Shaw, Tropical surfaces,

work page 2013

[11] [11]

Preprint available at arXiv:1506.07407.↑1 [She85] Allen Dudley Shepard, A cellular description of the derived category of a stratiﬁed space, Ph.D. Thesis,

work page internal anchor Pith review Pith/arXiv arXiv

[12] [12]

Swan, The theory of sheaves , Chicago lectures in mathematics, The University of Chicago Press, Chicago and London, 1964.↑16 [Wei94] Charles A

Available at https://epub.uni-regensburg.de/36262/.↑3 [Swa64] R.G. Swan, The theory of sheaves , Chicago lectures in mathematics, The University of Chicago Press, Chicago and London, 1964.↑16 [Wei94] Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge,

work page 1964

[13] [13]

Gökova Geom

MR1269324↑24 [Zha13] Ilia Zharkov, The Orlik-Solomon algebra and the Bergman fan of a matroid, J. Gökova Geom. Topol. GGT 7 (2013), 25–31. MR3153919↑8 E-mail address: andreas.gross@colostate.edu E-mail address: farbod@math.ku.dk, farbod@uw.edu

work page 2013