pith. sign in

arxiv: 1907.09416 · v1 · pith:7ESYBXQGnew · submitted 2019-07-22 · 🧮 math.CT · math.AT

Functors on Posets Left Kan Extend to Cosheaves: an Erratum

Justin Curry This is my paper

Pith reviewed 2026-05-24 17:37 UTC · model grok-4.3

classification 🧮 math.CT math.AT
keywords cosheavesleft Kan extensionsposetsdown-setserratumcocomplete categoriescategory theory
0
0 comments X

The pith

Functors from posets to cocomplete categories left Kan extend along the down-set embedding to cosheaves.

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

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.

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.

What carries the argument

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

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.

Where Pith is reading between the lines

These are 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.

Load-bearing premise

The codomain category must have all colimits.

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.

Figures

Figures reproduced from arXiv: 1907.09416 by Justin Curry.

Figure 1
Figure 1. Figure 1: A counter example to the statement that the cosheaf property is inher￾ited 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… view at source ↗
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.

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 / 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)
  1. [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.
  2. [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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 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.

axioms (1)
  • domain assumption The codomain category is cocomplete.
    Explicitly required in the abstract for the left Kan extension to be a cosheaf.

pith-pipeline@v0.9.0 · 5598 in / 1206 out tokens · 31405 ms · 2026-05-24T17:37:48.238713+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

  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

  9. [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

    work page 2018

  10. [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

    work page doi:10.1016/j.jpaa.2007.06.005 2008

  11. [11]

    Category theory in context

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

    work page 2017

  12. [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

    work page 2013

  13. [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)

    work page 1986

  14. [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

    work page 2016

  15. [15]

    The Stacks project

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

    work page 2019

  16. [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

    work page 2001

  17. [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

    work page 1991

  18. [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

    work page 1962

  19. [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

    work page 1962

  20. [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

    work page 1963