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arxiv: 2606.10553 · v1 · pith:UXERD6PDnew · submitted 2026-06-09 · 🧮 math.CT · math.AG

Kernel theorems for rigidly-compactly generated infty-categories

Giovanni Rossanigo This is my paper

Pith reviewed 2026-06-27 11:07 UTC · model grok-4.3

classification 🧮 math.CT math.AG
keywords representabilityrigidly-compactly generated ∞-categoriesperfect objectscoherent objectsGrothendieck dualityE∞-ring spectraspectral algebraic spaces
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The pith

Contravariant linear functionals out of perfect objects are represented by (pseudo)-coherent objects in rigidly-compactly generated ∞-categories.

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

The paper establishes two representability results for rigidly-compactly generated ∞-categories. The first shows that contravariant linear functionals from perfect objects to (pseudo)-coherent objects correspond to (pseudo)-coherent objects themselves. The second shows that covariant functionals from coherent objects to coherent objects correspond to perfect objects. These statements rest on the interplay of three finiteness conditions and apply to E∞-ring spectra, quasi-proper maps of schemes, and spectral algebraic spaces while recasting Grothendieck duality via internal left adjoints.

Core claim

In rigidly-compactly generated ∞-categories, contravariant linear functionals out of the category of perfect objects valued in (pseudo)-coherent objects are represented by (pseudo)-coherent objects, while covariant functionals out of coherent objects valued in coherent objects are represented by perfect objects.

What carries the argument

The interaction between compactness, dualizability, and coherence in presentable stable categories.

If this is right

  • The theorems apply to E∞-ring spectra.
  • The theorems apply to quasi-proper maps of quasi-compact quasi-separated schemes.
  • The theorems apply to certain spectral algebraic spaces.
  • Grothendieck duality admits a reformulation in terms of internal left adjoints.

Where Pith is reading between the lines

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

  • The same representability pattern could be checked in other presentable stable categories satisfying the three finiteness conditions.
  • The internal-left-adjoint reformulation of Grothendieck duality may simplify calculations involving dualizing objects in algebraic geometry.
  • Analogous kernel theorems might exist when compactness is replaced by other finiteness notions in ∞-categories.

Load-bearing premise

The three notions of finiteness interact in the required way inside rigidly-compactly generated ∞-categories.

What would settle it

An explicit rigidly-compactly generated ∞-category together with a linear functional that cannot be represented by the predicted (pseudo)-coherent or perfect object.

Figures

Figures reproduced from arXiv: 2606.10553 by Giovanni Rossanigo.

Figure 1
Figure 1. Figure 1: Empor, Vasilij Kandinskij, 1929. 1 arXiv:2606.10553v1 [math.CT] 9 Jun 2026 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
read the original abstract

We prove two representability results for rigidly-compactly generated $\infty$-categories and functors between them. The first one represents contravariant linear functionals out of a category of perfect objects with values in a category of (pseudo)-coherent objects in terms of (pseudo)-coherent objects. The second one represents covariant functionals out of coherent objects with values in a category of coherent objects in terms of perfect objects. The techniques used belong to the realm of "functional analysis" of presentable stable categories and ultimately depend on the interaction between three notion of finiteness, namely compactness, dualizability and coherence. These results apply to $\mathbb{E}_\infty$-ring spectra, quasi-proper maps of quasi-compact quasi-separated schemes and certain spectral algebraic spaces. We also reformulate Grothendieck duality in terms of internal left adjoints.

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 / 4 minor

Summary. The paper proves two representability results (kernel theorems) for rigidly-compactly generated ∞-categories and functors between them. The first represents contravariant linear functionals out of the category of perfect objects, valued in (pseudo)-coherent objects, in terms of (pseudo)-coherent objects. The second represents covariant functionals out of coherent objects, valued in coherent objects, in terms of perfect objects. The arguments rely on techniques from the 'functional analysis' of presentable stable ∞-categories and the interaction of three notions of finiteness: compactness, dualizability, and coherence. Applications are given to E_∞-ring spectra, quasi-proper maps of qcqs schemes, and certain spectral algebraic spaces; Grothendieck duality is also reformulated in terms of internal left adjoints.

Significance. If the results hold, they supply useful representability theorems that extend classical duality and representability statements from algebraic geometry and stable homotopy theory into the setting of rigidly-compactly generated ∞-categories. The reformulation of Grothendieck duality and the explicit applications to ring spectra and schemes are concrete strengths. The work is grounded in prior notions of finiteness rather than ad-hoc constructions.

minor comments (4)
  1. §1 (Introduction): the precise definition of 'rigidly-compactly generated' is referenced to prior work but not restated; a short self-contained reminder would help readers who are not experts in the cited papers.
  2. §3.2, Definition 3.4: the notation for the (pseudo)-coherent objects is introduced without an explicit comparison to the classical coherent objects in the non-∞ setting; a remark clarifying the relationship would improve readability.
  3. Theorem 4.1 and Theorem 5.3: the statements are clear, but the proofs would benefit from an explicit pointer to the lemma that encodes the key interaction between compactness and dualizability (currently only implicit in the argument flow).
  4. References: several citations to Lurie's Higher Algebra and Spectral Algebraic Geometry are given by page number only; adding theorem or proposition numbers would make the dependencies easier to verify.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of the manuscript. The recommendation for minor revision is noted, and we appreciate the recognition of the results' utility in extending classical duality statements to the setting of rigidly-compactly generated ∞-categories.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper states two representability theorems for rigidly-compactly generated ∞-categories that rest on the interaction of compactness, dualizability and coherence. No equations, fitted parameters, self-definitional reductions or load-bearing self-citations are exhibited in the provided text that would make any claimed result equivalent to its inputs by construction. The work is presented as a proof relying on established notions of finiteness, with applications to ring spectra and schemes; the central claims therefore remain independent of the inputs they derive from.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; the paper operates inside the established framework of presentable stable ∞-categories and therefore inherits all background axioms of that theory. No free parameters or new entities are mentioned.

axioms (1)
  • standard math Standard axioms and definitions of presentable stable ∞-categories, compactness, dualizability, and coherence as developed in prior literature.
    The abstract explicitly invokes these three notions of finiteness and the setting of rigidly-compactly generated ∞-categories.

pith-pipeline@v0.9.1-grok · 5661 in / 1441 out tokens · 54859 ms · 2026-06-27T11:07:45.296493+00:00 · methodology

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Reference graph

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