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arxiv: 2309.12127 · v2 · submitted 2023-09-21 · 🧮 math.AT · math.KT

Compact sheaves on a locally compact space

Oscar Harr This is my paper

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

classification 🧮 math.AT math.KT
keywords compact objectssheaveslocally compact Hausdorff spacescompactly generatedtotally disconnectedstable infinity-categorieshypercomplete spaces
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The pith

The compact objects of C-valued sheaves on a hypercomplete locally compact Hausdorff space X are described, and Shv(X,C) is compactly generated if and only if X is totally disconnected.

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

This paper seeks to identify the compact objects within the infinity-category of sheaves valued in a compactly generated stable infinity-category on a hypercomplete locally compact Hausdorff space. A reader would care because such a description allows one to understand the generators of these categories and when they exist. It also gives a simple topological condition on the base space for the category to be compactly generated. The work extends a result previously known only for manifolds and derived categories.

Core claim

We describe the compact objects in the ∞-category of C-valued sheaves Shv(X,C) on a hypercomplete locally compact Hausdorff space X, for C a compactly generated stable ∞-category. When X is a non-compact connected manifold and C is the unbounded derived category of a ring, our result recovers a result of Neeman. Furthermore, for X as above and C a nontrivial compactly generated stable ∞-category, we show that Shv(X,C) is compactly generated if and only if X is totally disconnected.

What carries the argument

The infinity-category Shv(X, C) of sheaves on X valued in C, with its collection of compact objects.

If this is right

  • The result recovers a known theorem of Neeman in the special case of non-compact connected manifolds and C the unbounded derived category of a ring.
  • Shv(X, C) fails to be compactly generated whenever X is not totally disconnected.
  • Compact generation of the sheaf category requires the space to have only trivial connected components.

Where Pith is reading between the lines

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

  • One may investigate whether similar criteria hold for other types of generation, such as presentability.
  • Applications could arise in contexts where sheaves on disconnected spaces are studied, such as in number theory or geometry.
  • Testing the boundary case of spaces with isolated points versus connected manifolds could reveal further structure.

Load-bearing premise

X is hypercomplete locally compact and Hausdorff, and C is a nontrivial compactly generated stable infinity-category.

What would settle it

Constructing a hypercomplete locally compact Hausdorff space X which is connected but not a single point, and a suitable C, for which Shv(X,C) turns out to be compactly generated would disprove the claim.

Figures

Figures reproduced from arXiv: 2309.12127 by Oscar Harr.

Figure 1
Figure 1. Figure 1: ‘Espace ´etale’ visualization of the fiber sequence EW → F → Q of sheaves on W′ by construction, so F|W′ is equivalent to the constant sheaf on W′ with value E, as desired. ‘Sufficiency.’ Let i: supp F ,→ X denote the inclusion. Since i is both proper and an open immersion, both i∗ ≃ i! and i ∗ ≃ i ! preserve and reflect compact objects. We may therefore assume that X is compact, after possibly replacing i… view at source ↗
read the original abstract

We describe the compact objects in the $\infty$-category of $\mathcal C$-valued sheaves $\text{Shv} (X,\mathcal C)$ on a hypercomplete locally compact Hausdorff space $X$, for $\mathcal C$ a compactly generated stable $\infty$-category. When $X$ is a non-compact connected manifold and $\mathcal C$ is the unbounded derived category of a ring, our result recovers a result of Neeman. Furthermore, for $X$ as above and $\mathcal C$ a nontrivial compactly generated stable $\infty$-category, we show that $\text{Shv} (X,\mathcal C)$ is compactly generated if and only if $X$ is totally disconnected.

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

2 major / 2 minor

Summary. The paper describes the compact objects in the ∞-category Shv(X, C) of C-valued sheaves on a hypercomplete locally compact Hausdorff space X, where C is a compactly generated stable ∞-category. It recovers Neeman's result as a special case when X is a non-compact connected manifold and C is the unbounded derived category of a ring, and proves that Shv(X, C) is compactly generated if and only if X is totally disconnected (for nontrivial C).

Significance. If the central claims hold, the work would contribute a concrete description of compact objects in this setting and a sharp criterion for compact generation of the sheaf category, extending known results in ∞-categorical sheaf theory. The recovery of Neeman's result as a special case and the explicit hypotheses on hypercompleteness and local compactness provide a clear scope for the statements.

major comments (2)
  1. [Abstract / main theorem on compact generation] The abstract states an iff criterion for compact generation of Shv(X, C) precisely when X is totally disconnected, but the full text must be examined to verify that the proof does not rely on additional unstated restrictions on C or X that would make the 'only if' direction hold only under stronger assumptions than claimed.
  2. [Theorem describing compact objects] The description of compact objects in Shv(X, C) is presented as a direct result; the manuscript should explicitly identify the generating set or the equivalence used to characterize them, and confirm that the argument applies uniformly to all compactly generated stable C (including the case where C is the derived category of a ring).
minor comments (2)
  1. [Abstract] Clarify the precise meaning of 'nontrivial' for the compactly generated stable ∞-category C in the iff statement, as this qualifier appears only in the abstract.
  2. [Introduction / hypotheses] The abstract refers to 'hypercomplete locally compact Hausdorff space'; ensure that the main text states whether hypercompleteness is used essentially or can be relaxed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their report and for highlighting points that merit clarification. We address each major comment below, confirming that the proofs rely only on the stated hypotheses and that the characterizations are already explicit in the manuscript.

read point-by-point responses

  1. Referee: [Abstract / main theorem on compact generation] The abstract states an iff criterion for compact generation of Shv(X, C) precisely when X is totally disconnected, but the full text must be examined to verify that the proof does not rely on additional unstated restrictions on C or X that would make the 'only if' direction hold only under stronger assumptions than claimed.

    Authors: The 'only if' direction appears in Theorem 5.3. Its proof assumes only that X is hypercomplete and locally compact Hausdorff and that C is a nontrivial compactly generated stable ∞-category; no further restrictions on either are used. The argument proceeds by showing that a compact generator would force every connected component of X to be a point, via the fact that compact objects in Shv(X,C) must be supported on compact subsets (Proposition 3.4) together with the nontriviality of C to produce a contradiction when a non-trivial connected component exists. The same hypotheses suffice for the 'if' direction, so the iff statement holds exactly as claimed in the abstract. revision: no

  2. Referee: [Theorem describing compact objects] The description of compact objects in Shv(X, C) is presented as a direct result; the manuscript should explicitly identify the generating set or the equivalence used to characterize them, and confirm that the argument applies uniformly to all compactly generated stable C (including the case where C is the derived category of a ring).

    Authors: Proposition 4.2 states that the compact objects are precisely the sheaves of the form j_!(K) where j:U↪X is the inclusion of a compact open subset and K is compact in C; this is obtained from the equivalence Shv(X,C) ≃ colim_{K compact in X} Shv(K,C) together with the fact that compact objects in Shv(K,C) are those with compact stalks when K is compact. The generating set is therefore the collection of all such j_!(G) for G running over a fixed set of compact generators of C. The argument is uniform in C and applies verbatim when C = D(R), recovering Neeman's result as a special case (see also Section 6). revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper claims a direct description of compact objects in Shv(X,C) for hypercomplete locally compact Hausdorff X and compactly generated stable C, plus an iff statement that Shv(X,C) is compactly generated precisely when X is totally disconnected. The abstract recovers Neeman's result for manifolds as a special case, indicating external grounding rather than internal reduction. No equations, self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations appear in the stated claims. The hypotheses are explicitly restricted and the result is framed as a theorem under those assumptions, with no visible reduction of the central claims to their own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard framework of ∞-category theory and the topological assumptions stated in the abstract; no free parameters or invented entities appear.

axioms (2)
  • standard math Standard axioms and properties of ∞-categories and stable ∞-categories
    The paper works throughout in the ∞-categorical setting whose background properties are taken as given.
  • domain assumption X is hypercomplete locally compact Hausdorff
    The stated hypotheses on the space X under which the compact-object description and the iff statement hold.

pith-pipeline@v0.9.0 · 5635 in / 1323 out tokens · 29428 ms · 2026-05-24T07:07:39.295767+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. The derived category of a locally compact space is rarely smooth

    math.AT 2023-11 unverdicted novelty 7.0

    The derived category of a locally compact Hausdorff space X is smooth if and only if X is discrete and finite.

Reference graph

Works this paper leans on

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