Morphisms and comorphisms of sites II -- Distributors of sites

Axel Osmond; Olivia Caramello

REVIEW 3 major objections 4 minor 17 references

Morphisms and comorphisms of sites II -- Distributors of sites

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Every geometric morphism between sheaf topoi is induced, uniquely up to isomorphism, by a continuous distributor of sites.

desk verdict Useful unification of morphisms and comorphisms of sites via distributors; central theorem likely correct but Lemma 2.1.13 has a gap as stated. read the letter →

arxiv 2507.20932 v1 pith:DZ4A3R5U submitted 2025-07-28 math.CT

Axel Osmond , Olivia Caramello This is my paper · ORCID

classification math.CT MSC 18B2518F10

keywords distributorsofsitesgeometricmorphismssheaftopoicover-flatnesscover-distributivitycomorphismsproarrowequipments

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

This paper introduces distributors of sites, a single kind of arrow between sites that contains both morphisms and comorphisms of sites as special cases. The central result is that, for small generated sites, continuous distributors of sites are in equivalence with geometric morphisms between the associated sheaf topoi: every geometric morphism is induced by a continuous distributor, uniquely up to isomorphism. The authors also show that the bicategory of sites and distributors of sites has several equipment-like features, including gluing of distributors, and that the continuous variant is the Kleisli bicategory of the sheafification pseudomonad. A sympathetic reader should care because the result gives a site-level, relational calculus for all morphisms of sheaf topoi, not only those coming from functors.

What carries the argument

The central object is a distributor of sites: a distributor $H : C \looparrowright D$, that is, a functor $D^{\mathrm{op}} \times C \to \mathrm{Set}$ thought of as heteromorphisms, equipped with two site-relative conditions. Cover-distributivity says that pulling back the image $H[S]$ of any $J$-covering sieve $S$ along any heteromorphism $x \in H(d,c)$ gives a $K$-covering sieve on $d$; $K$-flatness is the coverage-relative version of flatness, saying that $a_K \widehat{H} : C \to \widehat{D}_K$ is a flat functor, equivalently that the sheafified extension $a_K \mathrm{lext}_H i_J$ is lex. Continuity asks that $\widehat{H}$ take values in sheaves. The argument is carried by the composite $\mathrm{Sh}(H)^* = a_K \mathrm{lext}_H i_J$, which sends a distributor to the inverse image of a geometric morphism; compatibility of this construction with composition is what produces the equivalence.

What would settle it

Test Lemma 2.1.13 on a distributor that is $K$-flat but not cover-distributing, for example one induced by a flat but not cover-preserving functor: if $a_K \mathrm{lext}_H i_J$ is not lex, then the lemma as stated is false. To test Theorem 2.2.19, look for two continuous distributors between the same sites that induce isomorphic geometric morphisms but are not themselves isomorphic; the theorem predicts none exist, since the equivalence is categorical.

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Extended reading notes

Core claim

The paper's main discovery is Theorem 2.2.19: for any small generated sites $(C,J)$ and $(D,K)$ there is an equivalence of categories $\mathrm{Top}[\widehat{D}_K,\widehat{C}_J] \simeq \mathrm{SitW}_{\mathrm{Cont}}[(C,J),(D,K)]$. In plain terms, the geometric morphisms between the sheaf topoi are classified by the continuous distributors of sites between the sites themselves. A distributor of sites is a distributor $H : C \looparrowright D$ that is cover-distributing from $J$ to $K$ and $K$-flat, and it is continuous when the induced functor $\widehat{H} : C \to \widehat{D}$ lands in the sheaf topos $\widehat{D}_K$. The equivalence makes every geometric morphism $f$ arise as $\mathrm{Sh}(H_f)$ from the continuous distributor $H_f(d,c) = f^*(a_J \yo_c)(d)$, and it recovers the classical picture: representable distributors $D[1,f]$ give morphisms of sites, while corepresentable distributors $C[f,1]$ give comorphisms of sites.

Load-bearing premise

The load-bearing premise is that $K$-flatness of a distributor is equivalent to lexness of its sheafified extension $a_K \mathrm{lext}_H i_J$; the proof of Lemma 2.1.13 silently assumes the distributor is cover-distributing, a hypothesis absent from the lemma's statement but satisfied by the distributors of sites to which the main theorem applies.

Editorial extensions

If this is right

Morphisms of sites and comorphisms of sites are both recovered as distributors of sites: $D[1,f]$ for a morphism $f$ and $C[f,1]$ for a comorphism $f$ (Theorem 2.2.23).
Every geometric morphism between sheaf topoi admits a site-level presentation as a continuous distributor, even when no functorial lift exists between the fixed sites (Corollary 2.2.20).
Continuous distributors of sites form a bicategory $\mathrm{SitW}_{\mathrm{Cont}}$ with modified units and composition, and $\mathrm{Sh}$ factors through it; this bicategory is the Kleisli bicategory of the sheafification pseudomonad (Proposition 2.2.18 and 2.3.1).
A distributor of sites induces a geometric surjection exactly when it is cover-testing, a heteromorphic analogue of cover-reflecting (Proposition 2.2.27).
Distributors of sites can be glued into a site whose two inclusions are simultaneously morphisms and comorphisms of sites, giving cotabulators universal on both sides (Proposition 2.3.5).

Reading between the lines

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

Editorial inference: since the equivalence is phrased entirely in terms of sites and distributors, it supplies a calculus in which a geometric morphism can be manipulated as a relation of heteromorphisms; questions such as whether a morphism of topoi preserves a given logical structure could become concrete sieve conditions on the sites.
Editorial inference: the analogy drawn with join-approximable relations suggests a broader pattern, namely that distributors of sites may provide Stone-type dualities for other doctrines; this avenue is left open by the paper.
Editorial inference: because continuous distributors do not compose inside $\mathrm{SitW}$, the paper effectively constructs a 2-localization of the bicategory of sites and distributors; testing whether this localization has all the structure of a proarrow equipment once right adjoints are allowed in a suitably enlarged category would be a natural next step.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear Colleague,

The short version: this is a solid topos theory paper that introduces a genuinely useful notion, but the proof of Lemma 2.1.13 is incomplete as written. I think the main theorem survives, but a referee should ask for a fix.

What's new: the definition of distributor of sites (cover-distributing + K-flat) and the adjunction/equivalence between continuous distributors and geometric morphisms. That equivalence, Theorem 2.2.19, is a real result and unifies the two kinds of maps between sites. The paper also handles the subtlety that continuous distributors don't compose, and proposes a modified bicategory SitW_Cont. The gluing/equipment discussion at the end is a nice bonus, though it's more exploratory.

What's good: the authors are pretty honest about the relation to prior work. Proposition 2.2.9 connects their notion to Johnstone-Wraith's bidense continuity, which keeps the novelty claim calibrated. The hard direction, that every geometric morphism yields a distributor of sites, is grounded in Diaconescu's theorem rather than being engineered by definition. That's the right kind of argument.

The soft spot is Lemma 2.1.13. It states an equivalence for an arbitrary distributor H, but the proof says 'since H is in particular cover distributive' — an assumption that appears in neither the statement nor the definition of K-flatness. This isn't cosmetic: without cover-distributivity, aK bH need not send J-covers to epimorphic families, and the composite aKlextH iJ need not factor through aJ. So the lemma as stated is false or at least unproved. That said, the applications in Proposition 2.2.11 and Theorem 2.2.19 only use the lemma for distributors of sites, which are cover-distributing by definition. So the central claim is probably repairable by adding 'H cover-distributing' to the lemma. A referee should verify the proof goes through with that hypothesis.

Other quibbles: the abstract and introduction say 'sites' where the theorem needs 'small generated' sites; that's an overstatement. Some proofs, like Proposition 2.2.9, are sketches rather than full arguments. But these are minor relative to the main gap.

Bottom line: if you work in topos theory or Caramello's bridge program, this is worth reading and citing. It deserves a serious referee — the main idea is good and the flaw is localized. I'd recommend sending it to review with a request to fix Lemma 2.1.13 and tighten the small-generated hypotheses.

Best,

Referee Report

3 major / 4 minor

Summary. The paper introduces a notion of 'distributor of sites' between two sites (C,J) and (D,K), defined as a distributor H : C ↬ D that is cover-distributing from J to K and K-flat. It shows that such distributors induce geometric morphisms between the associated sheaf topoi, and that every geometric morphism arises from such a distributor (Proposition 2.2.11). A further 'continuity' condition on distributors (bH taking values in the sheaf topos) is introduced, and the paper claims in Theorem 2.2.19 an equivalence of categories Top[ bDK, bCJ ] ≃ SitW_Cont[(C,J),(D,K)], refining the adjunction of Proposition 2.2.11. The paper also discusses a bicategory SitW of sites and distributors of sites, its 'equipment-like' properties, gluing constructions, and connections to join-approximable relations and toposes.

Significance. If the main equivalence (Theorem 2.2.19) is correct, this is a substantial contribution: it provides a single profunctorial notion that generalizes both morphisms and comorphisms of sites, and it gives a classification of all geometric morphisms between sheaf topoi in terms of continuous distributors. The approach is natural and builds on standard tools (Diaconescu's theorem, flatness criteria, sheafification), and the paper makes a plausible case that the construction is pseudofunctorial. The paper also offers several useful auxiliary results, such as the gluing proposition (Proposition 2.3.5). However, the manuscript as written has a gap in a key lemma (Lemma 2.1.13) and, more importantly, the central theorem is stated without a proof. The overall framework is promising, but the paper needs substantial revision before the claims can be accepted.

major comments (3)

[§2.1, Lemma 2.1.13] The lemma states that for an arbitrary distributor H : C ↬ D, K-flatness is equivalent to flatness of aK bH and to lexness of aKlextH iJ, but the proof uses the hypothesis that H is cover-distributing ('since H is in particular cover distributive'). This hypothesis is not in the statement and is not part of Definition 2.1.11 of K-flatness. Moreover, the third equivalent condition (lexness of aKlextH iJ : bCJ → bDK) is only meaningful when aKlextH factors through aJ, which is exactly what cover-distributivity ensures via Lemma 2.1.10. Without that hypothesis, the composite aKlextH iJ may not be the induced inverse image functor on sheaves, so the stated equivalence is not correct. The fix is local: add 'H cover-distributing from J to K' to the lemma (or split the lemma into the unconditionally valid equivalence between K-flatness and flatness of aK bH, plus a separate statement for the lexness condition under cover-distributivity). Since all applications in §2.2 concern distributors of sites, which are cover-distributing by definition, the main theorem is likely repairable, but the proof as written is incomplete.
[§2.2, Theorem 2.2.19] The central result of the paper is stated without proof. The text preceding the theorem says that SitW_Cont[(C,J),(D,K)] is a reflective subcategory of SitW and that the adjunction of Proposition 2.2.11 restricts to an equivalence, but this is an assertion rather than a proof. A rigorous proof should show that for every continuous distributor H the unit H → H_{Sh(H)} is invertible (which follows if bH already lands in bDK), and that for every geometric morphism f the extracted distributor H_f is continuous and Sh(H_f) ≃ f. These steps are not written down, and because Theorem 2.2.19 is the main claim of the paper, this is a load-bearing gap.
[§2.2, Proposition 2.2.9] The proof of the converse direction ('Conversely suppose that H is Johnstone-Wraith continuous...') is only a sketch. In particular, the statement 'applying this fact to bidense morphisms into finite limits, aKlextH is exhibited as being lex, hence H to be K-flat' is not justified in detail, and the passage from Johnstone-Wraith continuity to K-flatness requires care because the definition of K-flatness involves arbitrary pairs and parallel pairs, not just finite limits of representables. Since this proposition is not used in the proof of Theorem 2.2.19, it is not a blocker, but it should be expanded for the paper to be self-contained.

minor comments (4)

[§2.2, Proposition 2.2.11] The notation ContJ[C, bDK] is used in the displayed sequence of isomorphisms but is never defined; please define it (presumably the category of functors C → bDK that send J-covers to epimorphic families).
[§2.2, Definition 2.2.12] In Definition 2.2.12, the phrase 'bH : C → bD takes values in the sheaf topos bDK' is slightly imprecise, since the codomain of bH is bD; it would be clearer to say that bH factors through the inclusion iK : bDK → bD.
[§2.2, Example 2.2.24] There is a typo: 'continuous ditributors' should be 'continuous distributors'.
[§2.1, Lemma 2.1.13, proof] The sentence 'by [13][VII.9 theorem 1] the functor aK bH is filtering if and only if it is flat, and this amounts for the induced aKlextH to being lex' is grammatically unclear and would benefit from being split into two sentences.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: Theorem 2.2.19 is grounded in Diaconescu's theorem and standard distributor facts; the continuity condition simply selects the essential image of the geometric-morphism correspondence. Minor non-circular issues: Lemma 2.1.13 uses an unstated cover-distributivity hypothesis, and self-citations ([7], [8]) are not load-bearing.

full rationale

The central equivalence Top[bDK,bCJ] ≃ SitW_Cont[(C,J),(D,K)] is not obtained by restating a definition. The hard direction, that every geometric morphism f : bDK → bCJ gives a continuous distributor of sites Hf, is derived through the Diaconescu correspondence: Hf(d,c) = f*(aJよc)(d), and the proof checks that aK bHf ≃ f*aJよC is flat and cover-preserving, with Lemma 2.1.13 and Lemma 2.1.10 translating these back to K-flatness and cover-distributivity. Conversely, the proof that a distributor of sites induces a geometric morphism uses the same external flatness criterion. The restriction to continuous distributors is explicitly acknowledged as selecting the image after sheafification: 'The hindrance to an equivalence is the problem that sheafification aK may identify distributors. This can be fixed by adding a condition of continuity' (Definition 2.2.12). This is a characterization of an essential image, not a prediction derived from its own conclusion. The paper's self-citations are not load-bearing for the main theorem: [8] supplies only double-categorical terminology, and [7] is used for the peripheral surjection criterion in Proposition 2.2.27. One genuine flaw appears in Lemma 2.1.13: its statement is for 'a distributor H : C ↬ D', but the proof begins 'by lemma 2.1.10 we know that, since H is in particular cover distributive', an assumption absent from both the lemma and Definition 2.1.11. This is a missing hypothesis rather than a circular step; the main theorem applies the lemma only to distributors of sites, which are cover-distributing by Definition 2.2.1, so the central claim is likely repairable. For these reasons, no circular reduction is exhibited.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The central claim rests on standard theorems in topos theory: Diaconescu's correspondence, flatness/filtering criteria, the existence of sheafification as a left exact localization, and the distributor calculus of Bénabou. The only domain-specific assumption is the small-generated site condition stated in the main theorems.

assumptions (5)

domain assumption Sites (C,J) and (D,K) are small and generated.
Invoked in Proposition 2.2.11 and Theorem 2.2.19; ensures sheafification and Diaconescu's theorem apply.
standard math Diaconescu's theorem: geometric morphisms Sh(D,K) -> Sh(C,J) correspond to J-continuous flat functors C -> Sh(D,K).
Used in the proof of Proposition 2.2.11 and Lemma 2.1.13, as stated at the end of the proof of Proposition 2.2.2.
standard math Mac Lane-Moerdijk flatness criteria (a functor is flat iff filtering; [13] VII.8/9).
Used in proof of Lemma 2.1.13 to translate K-flatness conditions into filteredness of aK bH.
standard math Sheafification aJ is a left exact localization, and J-bidense morphisms are the localized ones.
Used in Proposition 2.2.6 and 2.2.9 to connect cover-distributivity with bidense morphisms.
standard math Bénabou's distributor calculus, including the classifying equivalence Dist[C,D] ≃ [C,bD].
Foundation for all distributor computations in Section 1, especially the adjunction in Proposition 2.2.11.

invented entities (3)

Distributor of sites (cover-distributing and K-flat distributor) independent evidence
purpose: Unified notion generalizing morphisms and comorphisms of sites; its sheafified left extension induces a geometric morphism.
The definition is checkable; Proposition 2.2.9 equates it with the existing Johnstone-Wraith notion of continuity, and Theorem 2.2.19 gives a full correspondence with geometric morphisms.
Continuous distributor of sites independent evidence
purpose: Picks out the subcategory of distributors of sites equivalent to geometric morphisms, by requiring bH to land in the sheaf topos bDK.
Continuity is verified by checking that each bH(c) is a K-sheaf (Remark 2.2.13), an independent condition; Theorem 2.2.19 gives the equivalence.
Bicategory SitW (and its continuous variant SitW_Cont) independent evidence
purpose: Organizes sites, distributors of sites, and transformations; supports gluing and equipment-like structure.
Composition and units are established in Lemma 2.2.3 and 2.2.16; the gluing construction in Proposition 2.3.5 gives an independently checkable structure.

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Pith. "Pith review of Morphisms and comorphisms of sites II -- Distributors of sites." pith.science (2026). https://pith.science/paper/DZ4A3R5U

@misc{pith2026250720932,
  author       = {Pith},
  title        = {Pith review of: Morphisms and comorphisms of sites II -- Distributors of sites},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DZ4A3R5U}},
  note         = {Machine review of arXiv:2507.20932}
}

read the original abstract

We introduce a notion of distributor of sites, involving suited analogs of flatness and cover-preservation, and show that this notion jointly generalizes those of morphism and comorphism of sites. Given two sites, we exhibit an adjunction between the category of distributors of sites between them and the category of geometric morphisms between the associated sheaf topoi; this adjunction restricts to an equivalence between geometric morphisms and continuous distributors of sites. We finally discuss some equipment-like properties of the bicategory of sites and distributors of sites.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Oxford University Press, 1994

Samson Abramsky and Achim Jung.Domain theory. Oxford University Press, 1994

work page 1994

[2] [2]

Distributors at work

Jean Bénabou. “Distributors at work”. In: Lecture notes written by Thomas Streicher11 (2000)

work page 2000

[3] [3]

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Jean Bénabou. “Fibrations petites et localement petites”. In:CR Acad. Sc. Paris281 (1975), A831–A834

work page 1975

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Jean Bénabou. “Les distributeurs”. In:Université Catholique de Louvain, Institut de Mathé- matique Pure et Appliquée, rapport33 (1973), p. 519

work page 1973

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Handbook of Categorical Algebra: Volume 3, Sheaf Theory

Francis Borceux. Handbook of Categorical Algebra: Volume 3, Sheaf Theory. Vol. 3. Cam- bridge University Press, 1994

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[7] [7]

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Olivia Caramello. Denseness conditions, morphisms and equivalences of toposes. 2020. arXiv: 1906.08737 [math.CT]

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[8] [8]

Morphisms and comorphisms of sites I -- Double categories of sites

Olivia Caramello and Axel Osmond. “Morphisms and comorphisms of sites I – Double cate- gories of sites”. In:arXiv:2505.08766 (2025)

work page Pith review arXiv 2025