On fibred products of toposes

L\'eo Bartoli; Olivia Caramello

REVIEW 5 major objections 4 minor 9 references

On fibred products of toposes

T0 review · 5 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Pulling back a relative presheaf topos along any geometric morphism yields the relative presheaf topos on the inverse-image fibration.

desk verdict A genuinely new generalization of Giraud's pullback theorem to arbitrary fibrations, with a smart proof strategy but a heavy reliance on unpublished companions and several key lemmas left to the reader. read the letter →

arxiv 2509.07719 v1 pith:WVERZEZF submitted 2025-09-09 math.CT math.AG

Léo Bartoli , Olivia Caramello This is my paper · ORCID

classification math.CTmath.AG MSC 18B2518D3018F10

keywords relativetopostheoryfibrationsindexedcategoriespresheafGiraudpullbackoftoposesbipullbacketa-extension

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

The paper proves a base-change formula for relative toposes: pulling back a relative presheaf topos along any geometric morphism gives the relative presheaf topos on the inverse-image fibration. Earlier work had established this only for cartesian stacks, where finite limits can be checked fibrewise; the authors remove that restriction for arbitrary fibrations that are small relative to the base site. The payoff is that pullbacks of toposes presented by fibrations can be computed by base change on the indexing category, without explicit stackification or inverse-image calculations. The argument transposes morphisms of sites along the inverse/direct image adjunction and shows that this transposition preserves the property of being a morphism of sites.

What carries the argument

The load-bearing object is the $\eta$-extension with base change, a relative analogue of left Kan extension that sends a continuous functor between relative sites over different bases to a functor between their canonical stacks. Its right adjoint is defined through a pullback square, and the adjunction is what transfers morphisms of sites across the inverse/direct image adjunction. The other essential piece is the comparison functor between the inverse image of the canonical stack and the canonical stack of the inverse image, together with the image topology it induces; endowing the inverse image of the canonical stack with that topology makes the two sites Morita-equivalent, and this equivalence is what allows both directions of the transposition to preserve site morphisms.

What would settle it

Take a base site with one non-invertible arrow, let $\mathcal{C}$ be a non-cartesian indexed category over it, and let $f$ be the geometric morphism induced by a morphism of sites. Compute the $\eta$-extension of the structural map and its right adjoint directly: if the right adjoint does not send cartesian arrows to cartesian arrows, Proposition 3.1(i) fails and the proof of Proposition 4.16 collapses. At the level of the main claim, exhibit any small fibration $\mathcal{C}$ for which the two Hom-categories in Proposition 4.15 are not equivalent; that would be a concrete counterexample to the bipullback square.

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Extended reading notes

Core claim

Proposition 4.16 states that for a relative topos $f: \mathcal{F} \to \mathcal{E}$ and an $\mathcal{E}$-indexed category $\mathcal{C}$ whose fibration is small relative to the base site, the square formed by $\mathrm{Gir}(f^*\mathcal{C}) \to \mathrm{Gir}(\mathcal{C})$ and $\mathcal{F} \to \mathcal{E}$ is a bipullback of toposes. In words, the Giraud topos of the inverse-image fibration $f^*\mathcal{C}$ is the pullback of the Giraud topos of $\mathcal{C}$ along $f$. The paper reaches this by proving that transposition of morphisms of fibrations along the adjunction $f^* \dashv f_*$ restricts to an equivalence between categories of morphisms that are also morphisms of relative sites (Proposition 4.15). The key intermediate is a Morita equivalence, produced by an image topology, between the inverse image of the canonical stack and the canonical stack of the inverse image.

Load-bearing premise

The whole proof rests on an adjunction property whose proof is omitted in the paper: the $\eta$-extension with base change must be a left adjoint whose right adjoint preserves the fibred structure. If that property fails, the comparison functors used in Sections 4.3 through 4.5 cannot be constructed, and the bipullback theorem does not go through.

Editorial extensions

If this is right

For every relative presheaf topos presented by a fibration satisfying the smallness condition, pullback along any geometric morphism is the Giraud topos of the inverse-image fibration, so no stackification is needed to compute the pullback.
Giraud's pullback theorem, previously restricted to cartesian stacks, now covers arbitrary fibrations; fibre products in the bicategory of toposes can therefore be obtained from a wider class of site presentations.
Since inverse images along comorphisms of sites are computed by pullback of fibrations, the main theorem turns pullback of toposes into base change on the indexing category, a concrete operation.
The Morita equivalence between the inverse image of the canonical stack and the canonical stack of the inverse image supplies a canonical site-level presentation of the pullback topos that later arguments can use.

Reading between the lines

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

The transposition mechanism may extend from trivial relative sites to arbitrary relative site topologies, since the obstruction the paper overcomes concerns morphisms of sites rather than the particular choice of topology; a natural next step is to rerun Sections 4.3 through 4.5 with a general relative topology in place of the Giraud topology.
This suggests a general pullback-equals-inverse-image-fibration formula for toposes presented by fibrations with any topology preserved by the structural morphism; checking it against a known classifying-topos example would test whether the mechanism survives outside the presheaf case.
One could use Proposition 4.16 to compute pullbacks of classifying toposes of geometric theories directly from their syntactic fibrations, bypassing translation into internal categories; this would turn the theorem into a practical tool for geometric logic.

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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 headline is that this is a real contribution: it proves a genuinely new theorem — pullbacks of relative presheaf toposes for arbitrary fibrations, not just cartesian stacks — and the proof machinery (eta-extension with base change, the comparison indexed weak geometric morphism, the image topology trick) is substantial. Giraud's restriction to cartesian stacks was a genuine technical wall, and the paper identifies it clearly.

The good parts are real. Proposition 4.16 is new relative to Giraud and Pitts. The decision to work with comorphisms of sites to avoid explicit inverse-image computation is smart, and the Morita-equivalence between the inverse image of the canonical stack and the canonical stack of the inverse image is the right tool. The paper also honestly states where stackification is unnecessary.

Now the soft spots. The biggest one is not a fatal flaw but a serious verification burden: the two foundational propositions 3.1 and 3.3 are left to the reader, with proofs 'completely analogous' or 'entirely analogous' to results in the unpublished companion papers [BC23] and [BC25]. Since those papers are not available to the reader (and possibly not to the referee), the main proof is partly a black box. That is a real problem for a paper whose whole value is a new theorem.

On the stress-test note: I think its specific objection — that Sh(f*(η_C))* is undefined unless f*(η_C) is a morphism of sites — does not land, because continuous functors already induce topos-level adjunctions (Definition 2.9). So the composition identity can be written down. But there is a residual concern: the argument that Sh(^(f C)_t^*)* is an equivalence from surjectivity plus the composite equivalence is categorical and fine, but the diagram chase in Proposition 4.10 is dense, and the notation obscures the direction of units and transposes. A referee will need to check that carefully, and should have access to [BC23] and [BC25].

Also, the abstract omits the J-smallness hypothesis that appears in Definition 2.6 and is needed for the Giraud topos to exist properly. That is a presentation flaw, not a mathematical one.

Overall, the central idea holds up and the strategy is coherent. But the paper is not independently verifiable in its current form. It deserves a serious referee, with the expectation that the referee will request the companion papers.

Who should read it: specialists in relative topos theory and categorical logic. I'd bring it to a reading group only with a warning that it is heavy. I would cite it if I were working on relative topos pullbacks, but only after checking the companion papers.

Recommendation: accept for peer review, with a report that the authors need to either prove or precisely locate the foundational propositions and clarify the smallness condition.

Referee Report

5 major / 4 minor

Summary. The paper proves that the pullback (bipullback) of the Giraud topos of an arbitrary fibration along a geometric morphism is the Giraud topos of its inverse image. The main theorem is Proposition 4.16, which states that for a relative topos f:F→E and an E-indexed category C, the square Gir(f*C) → Gir(C), F → E is a bipullback of toposes. The proof introduces a notion of η-extension with base change, compares the canonical stack of a direct or inverse image with the direct or inverse image of the canonical stack, and uses image topologies to establish a Morita equivalence between the inverse image of the canonical stack and the canonical stack of the inverse image. The argument is a long chain that relies heavily on results from the authors' earlier manuscripts [BC23] and [BC25].

Significance. If correct, the paper substantially extends Giraud's and Diaconescu's pullback computations from cartesian stacks to arbitrary fibrations, which is a natural and valuable step in relative topos theory. The structural devices introduced—η-extensions with base change and the comparison functor ν—are original and likely to be reusable. The authors are transparent about deferring proofs to [BC23] and [BC25], but this means the present manuscript is largely a reduction to unpublished companion results. The main theorem is falsifiable and precise: for every E-indexed category C whose fibration is small relative to the base site, the square in Proposition 4.16 must satisfy the universal property of a bipullback. The paper would be a useful contribution once the deferred proofs are supplied or made publicly available.

major comments (5)

[Section 4.2–4.6] The main theorem depends essentially on unpublished black boxes: Theorem 3.15 of [BC23] (relative Diaconescu), Theorems 6.3.4 and 6.3.5 of [BC25] (indexed weak Diaconescu), and subsection 6.1 of [BC25] (equivalence between relative and indexed geometric morphisms). These results are used without proof in Propositions 4.9, 4.13, 4.15, and 4.16. For a journal publication, either full proofs must be included or the companion papers must be made available and shown to cover exactly the statements used. As it stands, the central claim of the paper is not verifiable from the manuscript alone.
[Section 3, Propositions 3.1 and 3.3] The proof of Proposition 3.1 is omitted entirely ('completely analogous to Proposition 3.4 [BC23] and left to the reader'), and the equivalence (i)⇔(iii) in Proposition 3.3 is also left to the reader. These propositions establish the adjunction properties of the η-extension with base change and the characterization of indexed weak geometric morphisms. Every later step in Section 4 uses these results, and the base-change case is not literally present in [BC23]. The omission is load-bearing and must be repaired by giving the proofs or by stating the base-change versions explicitly with detailed references.
[Proposition 4.12, proof] The proof that f*(η_C) is a morphism of sites is too compressed. The text first establishes continuity and then uses the factorization Sh(η_{f*C})* ≃ Sh(^(f C)*_t)* ∘ Sh(f*(η_C))* to deduce that Sh(^(f C)*_t)* is an equivalence and, finally, that Sh(f*(η_C))* is an equivalence. This reasoning is only valid if the reader accepts that Sh is defined for continuous functors as well as for morphisms of sites, and that an equivalence of inverse images forces the continuous functor to be a dense morphism of sites. The manuscript should state these conventions explicitly; as written, the argument risks a circularity and is not independently checkable.
[Proposition 4.16, statement] The statement of Proposition 4.16 omits the smallness hypothesis that is explicitly assumed in Section 4.2 ('for any fibration ... which is Jcan_E-small'). Without this hypothesis, the Giraud topos Gir(f*C) need not exist as a topos. The proposition should be restated with the smallness condition on C relative to the canonical topology of E.
[Proposition 4.16, bipullback universal property] The proof of Proposition 4.16 establishes, for an arbitrary relative topos g:G→F, an equivalence T op/E([fg],[Cp]) ≃ T op/F([g],[C_{p'}]). To conclude that the square is a bipullback in the bicategory of toposes, this equivalence must be 2-natural in g. The manuscript does not state or prove naturality. The authors should either verify pseudo-naturality of the equivalence or cite a general principle that makes it automatic from the construction.

minor comments (4)

[Throughout] The text contains many typographical errors, for example 'toposp' in Definition 2.8, 'categorie' in the introduction, and missing spaces such as 'Theorem2.3' and 'comorphism of sites' in Section 2.2. A careful proofreading pass is needed.
[Section 2.2, Definition 2.5] In the displayed description of the Giraud topology, the notation for the cartesian lifting family is ambiguous; it should be clarified that the family ((f_i,1)) is meant to be a family of cartesian arrows over a J-covering family.
[Section 3.1, Example 3.1] The description of the comparison indexed weak geometric morphism for étale toposes is helpful, but the sentence 'The terminal object of E/f*(F) is the identity morphism on f*(F)' should be rephrased: the terminal object is the pair (f*(F), id), and the unit f*f*(F)→F is not generally an isomorphism. The point is clear but the wording is imprecise.
[Section 4.5, proof of Proposition 4.13] The sentence 'It is also cartesian, and thus qualifies as a morphism of sites' is too terse. A morphism of fibrations that is cartesian in the fibrewise finite-limit sense is indeed a morphism of sites when both fibrations are cartesian and the codomain topology contains the Giraud topology, but this chain of implications should be spelled out for the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the bipullback theorem is not encoded in the definitions of f*C or Gir(-), and the cited Diaconescu-type theorems are general equivalences whose statements do not include the target result.

full rationale

The paper's central claim, Proposition 4.16, asserts a bipullback square of toposes for the Giraud topos of an arbitrary fibration. The derivation does not reduce to its inputs by construction. The inverse image f*C is independently defined via Lan_{f^op} (Proposition 2.6), and Gir(C) is defined by the Giraud topology (Definitions 2.5 and 2.8); neither definition contains the bipullback universal property. The proof establishes a transposition equivalence between site-morphism categories (Proposition 4.15), then applies relative Diaconescu's theorem to pass to geometric morphisms; this is a genuine derivation rather than a renaming of the conclusion. The heavy citations to [BC23] and [BC25] are self-citations, but they are used as general Diaconescu-type and local-fibration theorems whose stated scope does not include "pullback of a relative presheaf topos on an arbitrary fibration", so under the independence rule they do not raise the circularity score. The image topology in Proposition 4.11 is chosen so that ^(f_C)^t_* becomes a morphism of sites, but Proposition 4.12 then obtains the equivalence by cancellation from Sh(η_{f*C})_* ≃ Sh(^(f_C)^t_*)_* ∘ Sh(f*(η_C))_*; this is not assuming the conclusion. Two caveats, neither circular: Proposition 3.1's proof is omitted ('The proof of the proposition is completely analogous to that of Proposition 3.4 [BC23] and left to the reader'), and Proposition 4.12 establishes only continuity of f*(η_C) before using Sh(f*(η_C))_*; these are correctness and verification gaps, not definitional circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no numerical free parameters and no physically motivated new entities. It works within standard Grothendieck topos theory, but several load-bearing theorems are imported from the authors' own preprints [BC23] and [BC25], which are not machine-checked and whose proofs are not reproduced here. The new constructions, such as eta-extension with base change and the image topology, are mathematical tools rather than postulated entities.

assumptions (6)

standard math Fibrations over a base category C are equivalent to C-indexed categories (Proposition 2.1, cited from [CZ21]).
This equivalence is used throughout to present fibrations as indexed categories and to compute inverse images via Lan_{F^op}.
domain assumption All fibrations are assumed J-small, meaning the associated trivial relative site is small-generated (Definition 2.6).
The main theorem is stated for fibrations that are small relative to the base site; the abstract omits this hypothesis, though Section 4.2 and Definition 2.6 include it.
domain assumption The base topos E is presented by its canonical site (E, Jcan_E), so finite limits exist and the canonical stack is a cartesian fibration (Definition 2.12).
The proof repeatedly uses fibrewise finite limits and cartesianness of the canonical stack, which require the base topos to have finite limits.
ad hoc to paper Relative Diaconescu's theorem (Theorem 3.15 [BC23]) is used as a black box to convert morphisms of sites and fibrations into relative geometric morphisms.
This theorem is from the authors' own prior work and is not proved or machine-checked in the present paper, yet it is essential in Propositions 4.9, 4.15, and 4.16.
ad hoc to paper Indexed weak Diaconescu's theorem (Theorems 6.3.4 and 6.3.5 [BC25]) is used as a black box to relate indexed weak geometric morphisms to morphisms of fibrations.
This theorem is from the authors' own prior work, is invoked in Sections 4.2 and 4.4, and is not independently verified in this paper.
ad hoc to paper The equivalence between relative geometric morphisms and indexed geometric morphisms between canonical stacks (subsection 6.1 [BC25]) is used without proof in Proposition 4.16.
This equivalence is central to the final universal property argument and is inherited from the authors' earlier preprint [BC25].

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Cite this review

Pith. "Pith review of On fibred products of toposes." pith.science (2026). https://pith.science/paper/WVERZEZF

@misc{pith2026250907719,
  author       = {Pith},
  title        = {Pith review of: On fibred products of toposes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WVERZEZF}},
  note         = {Machine review of arXiv:2509.07719}
}

read the original abstract

In the setting of relative topos theory, we show that the pullback of a relative presheaf topos on an arbitrary fibration is the relative presheaf topos on its inverse image. To this end, we develop and exploit a notion of extension with base change of a morphism of sites along the canonical functor. This provides a tool to compare the canonical relative site of the direct (resp. inverse) image with the direct (resp. inverse) image of the canonical relative site: although these operations do not commute in general, we show that in the case of the direct image they are related by an indexed weak geometric morphism, while in the case of the inverse image they can be compared via a cartesian functor that induces a suitable topology making them Morita-equivalent.

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Works this paper leans on

9 extracted references · 3 canonical work pages

[1]

On morphisms of relative toposes, 2023

Léo Bartoli and Olivia Caramello. On morphisms of relative toposes, 2023

2023
[2]

Local fibrations and morphisms of relative toposes, 2025

Léo Bartoli and Olivia Caramello. Local fibrations and morphisms of relative toposes, 2025

work page 2025
[3]

Denseness conditions, morphisms and equivalences of toposes

Olivia Caramello. Denseness conditions, morphisms and equivalences of toposes. arXiv:math.CT/1906.08737v3, 2020

arXiv 1906
[4]

Fibred sites and existential toposes

Olivia Caramello. Fibred sites and existential toposes. \\ arXiv:math.AG/2212.11693, 2022

arXiv 2022
[5]

Morphisms and comorphisms of sites i -- double categories of sites, 2025

Olivia Caramello and Axel Osmond. Morphisms and comorphisms of sites i -- double categories of sites, 2025

work page 2025
[6]

Relative topos theory via stacks

Olivia Caramello and Riccardo Zanfa. Relative topos theory via stacks. arXiv:math.AG/2107.04417, 2021

arXiv 2021
[7]

Classifying topos

Jean Giraud. Classifying topos. In F. W. Lawvere, editor, Toposes, Algebraic Geometry and Logic , pages 43--56. Springer Berlin Heidelberg, 1972

1972
[8]

Johnstone

Peter T. Johnstone. Sketches of an elephant: a topos theory compendium. V ol. 1 and 2 , volume 43-44 of Oxford Logic Guides . Oxford University Press, Oxford, 2002

2002

Show all 9 references

[9]

On product and change of base for toposes

Andrews Pitts. On product and change of base for toposes. Cahiers de Topologie et G\'eom\'etrie Diff\'erentielle Cat\'egoriques , 26:43--61, 1985

1985

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

On morphisms of relative toposes, 2023

Léo Bartoli and Olivia Caramello. On morphisms of relative toposes, 2023

2023

[2] [2]

Local fibrations and morphisms of relative toposes, 2025

Léo Bartoli and Olivia Caramello. Local fibrations and morphisms of relative toposes, 2025

work page 2025

[3] [3]

Denseness conditions, morphisms and equivalences of toposes

Olivia Caramello. Denseness conditions, morphisms and equivalences of toposes. arXiv:math.CT/1906.08737v3, 2020

arXiv 1906

[4] [4]

Fibred sites and existential toposes

Olivia Caramello. Fibred sites and existential toposes. \\ arXiv:math.AG/2212.11693, 2022

arXiv 2022

[5] [5]

Morphisms and comorphisms of sites i -- double categories of sites, 2025

Olivia Caramello and Axel Osmond. Morphisms and comorphisms of sites i -- double categories of sites, 2025

work page 2025

[6] [6]

Relative topos theory via stacks

Olivia Caramello and Riccardo Zanfa. Relative topos theory via stacks. arXiv:math.AG/2107.04417, 2021

arXiv 2021

[7] [7]

Classifying topos

Jean Giraud. Classifying topos. In F. W. Lawvere, editor, Toposes, Algebraic Geometry and Logic , pages 43--56. Springer Berlin Heidelberg, 1972

1972

[8] [8]

Johnstone

Peter T. Johnstone. Sketches of an elephant: a topos theory compendium. V ol. 1 and 2 , volume 43-44 of Oxford Logic Guides . Oxford University Press, Oxford, 2002

2002