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arxiv: 2605.22499 · v1 · pith:BSHGGCQ3new · submitted 2026-05-21 · 🧮 math.AG · math.CT

A condensed proof of the pro-\'etale and \'etale exodromy theorems

Remy van Dobben de Bruyn This is my paper

Pith reviewed 2026-05-22 02:08 UTC · model grok-4.3

classification 🧮 math.AG math.CT
keywords exodromypro-etale sheavesetale sheavescondensed categoriesGalois categoryconstructible sheavesinfinity-categories
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The pith

Treating Gal(X) as a condensed category from the outset yields quick proofs of the pro-étale and étale exodromy theorems.

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

The paper aims to prove exodromy correspondences using a condensed perspective on the Galois category of a scheme. By starting with condensed categories, it delivers a self-contained proof for pro-étale sheaves of spaces. This proof then allows extraction of a similar correspondence for Postnikov complete étale sheaves. The approach also provides a new proof for constructible étale sheaves and applies more generally without qcqs assumptions.

Core claim

Viewing the profinite category Gal(X) as a condensed category establishes a correspondence that identifies pro-étale sheaves with continuous functors out of it, and this perspective extends to give an exodromy theorem for étale sheaves.

What carries the argument

The condensed category Gal(X), which carries the exodromy correspondence to recover the usual sheaf categories directly.

If this is right

  • The method eliminates the need for quasi-compact quasi-separated hypotheses on the scheme X.
  • It extends the exodromy to sheaves with coefficients in arbitrary infinity-categories.
  • It produces a version of the theorem that is kappa-condensed for kappa larger than the cardinality of the ring of functions on any affine open.
  • The condensed proof also recovers the constructible case as a corollary.

Where Pith is reading between the lines

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

  • This could link the exodromy to ultracategory structures in other contexts.
  • Similar condensed methods might simplify proofs in other parts of algebraic geometry involving profinite structures.
  • Checking the correspondence explicitly on a non-quasi-compact scheme would test the removal of restrictions.

Load-bearing premise

The condensed category on Gal(X) recovers the standard pro-étale and étale sheaf categories without requiring extra quasi-compactness or size bounds.

What would settle it

A specific scheme X and a pro-étale sheaf on it that cannot be represented as a continuous functor from the condensed Gal(X).

read the original abstract

The exodromy correspondence of Barwick, Glasman, and Haine computes constructible sheaves of spaces on a scheme $X$ as an $\infty$-category of continuous functors from the profinite category $\operatorname{Gal}(X)$. Viewing $\operatorname{Gal}(X)$ instead as a condensed category, this was extended by Wolf to an exodromy correspondence for pro-\'etale sheaves. Using the condensed perspective from the outset, we give a quick and self-contained proof of the pro-\'etale exodromy theorem. This is used to extract an exodromy theorem for (Postnikov complete) \'etale sheaves that does not yet appear in the literature, which is closely related to Lurie's work on ultracategories. Finally, we use this to give a new proof of the constructible \'etale exodromy correspondence of Barwick, Glasman, and Haine. Without additional effort, our method removes the qcqs hypotheses on the schemes, and gives versions for sheaves with coefficients in more general $\infty$-categories. Finally, we refine the methods to obtain a $\kappa$-condensed statement whenever $\kappa > \lvert \mathcal O_X(U) \rvert$ for every affine open $U \subseteq X$.

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

1 major / 2 minor

Summary. The manuscript claims to provide a quick and self-contained proof of the pro-étale exodromy theorem by adopting the condensed viewpoint from the beginning. It then extracts an exodromy theorem for Postnikov-complete étale sheaves, offers a new proof of the constructible étale exodromy correspondence of Barwick-Glasman-Haine, removes the quasi-compact quasi-separated hypotheses on the schemes, extends to sheaves with coefficients in more general ∞-categories, and refines the methods to obtain κ-condensed statements for κ larger than the cardinality of sections of the structure sheaf on affine opens.

Significance. If the derivations hold, the work is significant for simplifying the proofs of exodromy theorems and extending their applicability to arbitrary schemes without size restrictions and to broader coefficient categories. The self-contained nature starting from condensed categories is a strength, as is the connection to ultracategories. This could impact the study of constructible sheaves in algebraic geometry.

major comments (1)
  1. [Central construction] Central construction (pro-étale exodromy equivalence): the claim that treating Gal(X) as a condensed category from the outset recovers the usual pro-étale and étale sheaf categories without qcqs or size restrictions on X rests on the condensed profinite completion commuting with the relevant limits/colimits that define sheaves on the étale site. For non-qcqs schemes this includes commutation with infinite disjoint unions appearing in étale covers; the text invokes the universal property of condensed categories but does not explicitly verify the commutation in this setting.
minor comments (2)
  1. Notation for condensed categories and profinite completions could be introduced with a brief reminder of the relevant universal properties for readers coming from the classical exodromy literature.
  2. The refinement to κ-condensed statements is stated in the abstract but the precise dependence on |O_X(U)| is not cross-referenced to the main theorems.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the significance of the work and for the careful reading of the central construction. We address the concern below and have revised the manuscript to make the relevant commutation properties fully explicit.

read point-by-point responses

  1. Referee: [Central construction] Central construction (pro-étale exodromy equivalence): the claim that treating Gal(X) as a condensed category from the outset recovers the usual pro-étale and étale sheaf categories without qcqs or size restrictions on X rests on the condensed profinite completion commuting with the relevant limits/colimits that define sheaves on the étale site. For non-qcqs schemes this includes commutation with infinite disjoint unions appearing in étale covers; the text invokes the universal property of condensed categories but does not explicitly verify the commutation in this setting.

    Authors: We thank the referee for pinpointing this point. The argument in the manuscript proceeds from the universal property that the condensed profinite completion is the left adjoint to the forgetful functor from condensed categories to ordinary categories; as a left adjoint it automatically commutes with all colimits, including the infinite disjoint unions that may appear in étale covers of non-qcqs schemes. Because the pro-étale site is generated under colimits by profinite sets, this yields the identification of sheaf categories without any qcqs or cardinality hypotheses. We agree, however, that the original text did not spell out the commutation for infinite coproducts in the non-qcqs case. In the revised version we have inserted a short new paragraph (immediately after the statement of the universal property in Section 2) that records this fact explicitly, together with the observation that infinite coproducts of condensed sets are computed pointwise on extremally disconnected spaces. This makes the recovery of the usual sheaf categories fully rigorous in the stated generality. revision: yes

Circularity Check

0 steps flagged

Self-contained condensed derivation of exodromy theorems with no load-bearing reductions to inputs or self-citations

full rationale

The paper starts from the condensed viewpoint on Gal(X) to derive the pro-étale exodromy equivalence directly and then extracts the Postnikov-complete étale version. This is presented as self-contained and removes qcqs hypotheses without additional effort. No quoted equations or steps reduce by construction to fitted parameters, prior self-cited results, or renamings of known patterns. The central construction relies on the universal property of condensed categories applied to the profinite Galois category, which is independent of the target sheaf categories. This is a normal low-score outcome for a paper whose derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the existing framework of condensed mathematics, the prior exodromy results of Barwick-Glasman-Haine and Wolf, and standard axioms of ∞-category theory; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard axioms and limits of ∞-category theory as in Lurie's Higher Topos Theory
    The paper works throughout in the ∞-categorical setting and invokes Postnikov completeness.
  • domain assumption The profinite category Gal(X) can be promoted to a condensed category
    This is the key modeling choice stated in the abstract as the starting point for the proof.

pith-pipeline@v0.9.0 · 5766 in / 1433 out tokens · 57135 ms · 2026-05-22T02:08:54.914154+00:00 · methodology

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