Functors on Posets Left Kan Extend to Cosheaves: an Erratum

Justin Curry

Functors on Posets Left Kan Extend to Cosheaves: an Erratum

T0 review · 0 major / 2 minor · reviewed 2026-05-24 · grok-4.3

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Pith's one-line read Functors from posets to cocomplete categories left Kan extend along the down-set embedding to cosheaves.

desk verdict This erratum gives a clean self-contained proof of the standard fact that left Kan extensions along the down-set embedding produce cosheaves when the target category is cocomplete, fixing errors in the author's earlier work. read the letter →

arxiv 1907.09416 v1 pith:7ESYBXQG submitted 2019-07-22 math.CT math.AT

Justin Curry This is my paper · ORCID

classification math.CTmath.AT

keywords cosheavesleftKanextensionsposetsdown-setserratumcocompletecategoriescategorytheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper supplies a self-contained proof of the claim that any functor whose domain is a poset and whose codomain is a cocomplete category has a left Kan extension that is a cosheaf when taken along the embedding of the poset into its poset of down-sets. This statement corrects mistaken arguments that appeared in the author's earlier thesis and in a paper on dualities between sheaves and cosheaves. A reader would care because the result supplies an explicit, standard construction that turns data defined only on the poset into a cosheaf on a larger site. The proof works by direct verification that the Kan extension satisfies the cosheaf colimit-preservation condition.

What carries the argument

Left Kan extension of F along the inclusion of the poset P into its down-set poset.

What would settle it

A concrete functor from a poset into a category lacking some colimits whose left Kan extension along the down-set embedding fails the cosheaf colimit condition.

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Extended reading notes

Core claim

If F is a functor from a poset P to a cocomplete category C, then the left Kan extension of F along the embedding of P into the poset of down-sets of P is a cosheaf.

Load-bearing premise

The codomain category must have all colimits.

Editorial extensions

If this is right

Cosheaves on the down-set poset arise directly from arbitrary functors on the original poset.
The construction requires only that the target category be cocomplete.
The result supplies a foundation for exchanging cellular sheaves and cosheaves via duality.
Verification of the cosheaf property can now proceed without reference to the earlier flawed proofs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same construction may simplify explicit calculations of cosheaf homology groups in concrete examples.
One could check whether analogous left Kan extensions remain cosheaves when the site is enlarged beyond down-sets of posets.
The statement immediately yields a supply of cosheaves valued in any cocomplete category such as sets or vector spaces.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

This erratum fixes the proofs for a basic fact about cosheaves on posets. If F maps a poset to a cocomplete category, then its left Kan extension along the down-set embedding preserves colimits and is therefore a cosheaf. The earlier proofs in the thesis and the dualities paper had errors, so this note replaces them. The paper does well by giving an explicit, self-contained argument that relies only on the cocompleteness of the target. That hypothesis is precisely what lets the colimits in the Kan extension formula exist, and the cosheaf property then follows from the universal property of the free cocompletion. The result is not new. It is the standard way to extend a functor to a colimit-preserving one. The value lies in the correction and in having a clean reference. Soft spots are limited. The scope is narrow because it only addresses the author's own prior claims. No new examples or generalizations appear. The proof is presented as straightforward, and the stress-test note confirms it aligns with the expected construction, so no load-bearing issues stand out. This is useful for anyone citing the earlier papers on cellular sheaves and cosheaves, especially in algebraic topology or data analysis contexts. A reader looking for a reliable statement of this construction will find it here. I would bring this to a reading group only if the group is specifically working through cosheaf theory references. It deserves peer review as a short note to ensure the corrected proof is solid before it replaces the old ones in the literature.

Referee Report

0 major / 2 minor

Summary. The manuscript is an erratum providing a self-contained proof that for any functor F: P → C where P is a poset and C is cocomplete, the left Kan extension Lan_y F: Down(P) → C along the embedding y: P ↪ Down(P) into the poset of down-sets is a cosheaf (i.e., colimit-preserving).

Significance. The result is a standard fact equivalent to the universal property of the free cocompletion of a poset under colimits; the self-contained proof corrects errors in the author's prior thesis and article. This strengthens accessibility for readers working with cosheaves on posets without invoking the full machinery of Kan extensions or presheaf categories.

minor comments (2)

[Abstract] The abstract refers to 'the poset of down-sets' without explicitly noting that Down(P) carries the inclusion order making y order-preserving and dense.
[Proof section] In the proof, the colimit formula for Lan_y F should be cross-referenced to the standard expression in terms of down-sets to aid readers unfamiliar with the poset case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation to accept the manuscript. We are pleased that the self-contained nature of the proof is viewed as improving accessibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The paper presents a self-contained proof that for any functor F from a poset P to a cocomplete category C, the left Kan extension Lan_y F along the Yoneda embedding y: P → Down(P) is a cosheaf (i.e., colimit-preserving). This is the standard universal property of the free cocompletion of P under colimits; the cocompleteness assumption ensures the relevant colimits exist in the Kan formula, and the proof does not reduce any central claim to a fitted parameter, self-citation chain, or definitional tautology. The erratum explicitly replaces prior mistaken proofs with an independent derivation, confirming the result stands on its own without circular reduction.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Review is based solely on the abstract; no free parameters, invented entities, or non-standard axioms are visible. The claim rests on the standard definition of left Kan extension and the assumption that the codomain admits all colimits.

assumptions (1)

domain assumption The codomain category is cocomplete.
Explicitly required in the abstract for the left Kan extension to be a cosheaf.

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Cite this review

Pith. "Pith review of Functors on Posets Left Kan Extend to Cosheaves: an Erratum." pith.science (2026). https://pith.science/paper/7ESYBXQG

@misc{pith2026190709416,
  author       = {Pith},
  title        = {Pith review of: Functors on Posets Left Kan Extend to Cosheaves: an Erratum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7ESYBXQG}},
  note         = {Machine review of arXiv:1907.09416}
}

read the original abstract

In this note we give a self-contained proof of a fundamental statement in the study of cosheaves over a poset. Specifically, if a functor has domain a poset and co-domain a co-complete category, then the left Kan extension of that functor along the embedding of the domain poset into its poset of down-sets is a cosheaf. This proof is meant to replace the mistaken proofs published in the author's thesis and an article on dualities exchanging cellular sheaves and cosheaves.

Figures

Figures reproduced from arXiv: 1907.09416 by the authors.

**Figure 1.** A counter example to the statement that the cosheaf property is inherited by coarser covers. given by the indicated vertices. For the covers considered in our example, we need only use unions of open stars of cells. Near each open set we have indicated the value of a precosheaf F, valued in k-vector spaces. For our example, we have F(X) = k, F(V2) = k ‘ k – k 2 , and all other indicated open sets are also assigned … view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

Modules over posets: commutative and homological algebra
math.AC 2019-08 conditional novelty 8.0 of 10

Tame modules over arbitrary posets are proved to have finite encodings, fringe presentations, indicator resolutions, and primary decompositions, leading to proofs of two sheaf-theoretic conjectures.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · cited by 1 Pith paper

[1]

Whitney numbers of geometric lattices

Kenneth Bac lawski. “Whitney numbers of geometric lattices”. In: Advances in Mathe- matics 16.2 (1975), pp. 125 –138. issn: 0001-8708. doi: https://doi.org/10.1016/ 0001-8708(75)90145-0

work page 1975
[2]

Ephemeral persistence modules and distance comparison

Nicolas Berkouk and Fran¸ cois Petit. “Ephemeral persistence modules and distance comparison”. In: arXiv preprint arXiv:1902.09933 (2019)

work page arXiv 1902
[3]

Sheaves, Cosheaves and Applications

Justin Curry. “Sheaves, Cosheaves and Applications”. PhD thesis. University of Penn- sylvania, 2014

work page 2014
[4]

Dualities between cellular sheaves and cosheaves

Justin Michael Curry. “Dualities between cellular sheaves and cosheaves”. In: Journal of Pure and Applied Algebra 222.4 (2018), pp. 966–993

work page 2018
[5]

Topological hypercovers and 1-realizations

Daniel Dugger and Daniel C Isaksen. “Topological hypercovers and 1-realizations”. In: Mathematische Zeitschrift 246.4 (2004), pp. 667–689

work page 2004
[6]

Network codings and sheaf cohomology

Robert Ghrist and Yasuaki Hiraoka. “Network codings and sheaf cohomology”. In: Proc. NOLTA. 2011, pp. 266–269

work page 2011
[7]

Toward a Spectral Theory of Cellular Sheaves

Jakob Hansen and Robert Ghrist. “Toward a Spectral Theory of Cellular Sheaves”. In: arXiv preprint arXiv:1808.01513 (2018)

work page arXiv 2018
[8]

The Riemann-Hilbert problem for holonomic systems

Masaki Kashiwara. “The Riemann-Hilbert problem for holonomic systems”. In: Publi- cations of the Research Institute for Mathematical Sciences 20.2 (1984), pp. 319–365

work page 1984

Show all 20 references

[9]

Persistent homology and microlocal sheaf theory

Masaki Kashiwara and Pierre Schapira. “Persistent homology and microlocal sheaf theory”. In: Journal of Applied and Computational Topology 2.1-2 (2018), pp. 83–113. 10 REFERENCES

2018
[10]

On derived equivalences of categories of sheaves over ﬁnite posets

Seﬁ Ladkani. “On derived equivalences of categories of sheaves over ﬁnite posets”. In: Journal of Pure and Applied Algebra 212.2 (2008), pp. 435 –451. issn: 0022-4049. doi: https://doi.org/10.1016/j.jpaa.2007.06.005

2008 doi
[11]

Category theory in context

Emily Riehl. Category theory in context . Courier Dover Publications, 2017

2017
[12]

The Nyquist theorem for cellular sheaves

Michael Robinson. “The Nyquist theorem for cellular sheaves”. In: 10th international conference on Sampling Theory and Applications (SampTA 2013) . Bremen, Germany, July 2013, pp. 293–296

2013
[13]

A CELLULAR DESCRIPTION OF THE DERIVED CATE- GORY OF A STRATIFIED SPACE

Allen Dudley Shepard. “A CELLULAR DESCRIPTION OF THE DERIVED CATE- GORY OF A STRATIFIED SPACE.” In: (1986)

1986
[14]

Categoriﬁed Reeb Graphs

Vin de Silva, Elizabeth Munch, and Amit Patel. “Categoriﬁed Reeb Graphs”. In: Dis- crete & Computational Geometry (2016), pp. 1–53. issn: 1432-0444. doi: 10.1007/ s00454-016-9763-9

2016
[15]

The Stacks project

The Stacks project authors. The Stacks project. https://stacks.math.columbia.edu. Tag 009H or Section 6.30. 2019

2019
[16]

Sheaves on ﬁnite posets and modules over normal semigroup rings

Kohji Yanagawa. “Sheaves on ﬁnite posets and modules over normal semigroup rings”. In: Journal of Pure and Applied Algebra 161.3 (2001), pp. 341–366

2001
[17]

Cohomology of local sheaves on arrangement lattices

Sergey Yuzvinsky. “Cohomology of local sheaves on arrangement lattices”. In: Proceed- ings of the American Mathematical Society 112.4 (1991), pp. 1207–1217

1991
[18]

Dihomology: I. Relations Between Homology Theories

EC Zeeman. “Dihomology: I. Relations Between Homology Theories”. In: Proceedings of the London Mathematical Society 3.1 (1962), pp. 609–638

1962
[19]

Dihomology: II. The Spectral Theories of a Map

EC Zeeman. “Dihomology: II. The Spectral Theories of a Map”. In: Proceedings of the London Mathematical Society 3.1 (1962), pp. 639–689

1962
[20]

Dihomology III. A generalization of the Poincar´ e duality for manifolds

EC Zeeman. “Dihomology III. A generalization of the Poincar´ e duality for manifolds”. In: Proceedings of the London Mathematical Society 3.1 (1963), pp. 155–183

1963

Pith tools

Reviewed May 24, 2026 · model on record in the stance chip above.

[1] [1]

Whitney numbers of geometric lattices

Kenneth Bac lawski. “Whitney numbers of geometric lattices”. In: Advances in Mathe- matics 16.2 (1975), pp. 125 –138. issn: 0001-8708. doi: https://doi.org/10.1016/ 0001-8708(75)90145-0

work page 1975

[2] [2]

Ephemeral persistence modules and distance comparison

Nicolas Berkouk and Fran¸ cois Petit. “Ephemeral persistence modules and distance comparison”. In: arXiv preprint arXiv:1902.09933 (2019)

work page arXiv 1902

[3] [3]

Sheaves, Cosheaves and Applications

Justin Curry. “Sheaves, Cosheaves and Applications”. PhD thesis. University of Penn- sylvania, 2014

work page 2014

[4] [4]

Dualities between cellular sheaves and cosheaves

Justin Michael Curry. “Dualities between cellular sheaves and cosheaves”. In: Journal of Pure and Applied Algebra 222.4 (2018), pp. 966–993

work page 2018

[5] [5]

Topological hypercovers and 1-realizations

Daniel Dugger and Daniel C Isaksen. “Topological hypercovers and 1-realizations”. In: Mathematische Zeitschrift 246.4 (2004), pp. 667–689

work page 2004

[6] [6]

Network codings and sheaf cohomology

Robert Ghrist and Yasuaki Hiraoka. “Network codings and sheaf cohomology”. In: Proc. NOLTA. 2011, pp. 266–269

work page 2011

[7] [7]

Toward a Spectral Theory of Cellular Sheaves

Jakob Hansen and Robert Ghrist. “Toward a Spectral Theory of Cellular Sheaves”. In: arXiv preprint arXiv:1808.01513 (2018)

work page arXiv 2018

[8] [8]

The Riemann-Hilbert problem for holonomic systems

Masaki Kashiwara. “The Riemann-Hilbert problem for holonomic systems”. In: Publi- cations of the Research Institute for Mathematical Sciences 20.2 (1984), pp. 319–365

work page 1984