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REVIEW 3 major objections 4 minor 22 references

Sites and Grothendieck toposes: an introduction

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Topos theory unifies geometry and logic: spaces are sheaf categories, theories are classifying toposes.

desk verdict This is a textbook opening, not a research paper, and the only fully supplied chapter already contains a concrete adjunction error: fixed points and orbits are swapped in §I.8.b. read the letter →

arxiv 2508.21609 v1 pith:VDULOQ4M submitted 2025-08-29 math.CT math.AGmath.LO

Olivia Caramello , Laurent Lafforgue This is my paper · ORCID
classification math.CTmath.AGmath.LO MSC 18B2518F1018F2003G30
keywords GrothendiecktopossitesheafclassifyinggeometriclogicGiraudtheoremDiaconescuequivalenceMorita
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

Planned as a five-chapter introduction—with Chapter I and the full table of contents presented in this excerpt—the book argues that topos theory is the common language of geometry and mathematical logic. It starts from familiar material (groups and their actions, topological spaces, sheaves, categories) and moves to Grothendieck topologies and sites, then to the axiomatic characterization of toposes, their geometry, and finally to classifying toposes for geometric first-order theories. The thesis is reversible: a space can be read as a category of sheaves on a site, and a geometric theory can be read as a topos whose models are the morphisms into it. A reader with basic algebra, topology, and category theory is meant to finish with a working dictionary in which local and global, external and internal, syntactic and semantic viewpoints are translations of one another.

What carries the argument

The carrying mechanism is the site-to-topos passage and its logical reverse. A site is a small category with a Grothendieck topology (covering sieves); its topos of sheaves glues objects from those coverings. A geometric first-order theory builds a syntactic category—objects are formulas—with the syntactic topology; the sheaf topos on it is the classifying topos. The two directions are joined by an equivalence between geometric morphisms into a sheaf topos and flat continuous functors from the site (Diaconescu's equivalence). It makes models in any topos correspond to morphisms into the classifying topos, and lets any site-presented topos classify a geometric theory.

What would settle it

Open the completed Chapter V: if the classifying topos is constructed for every geometric first-order theory via the syntactic category and syntactic topology, and the proof of universality uses the categorical description of morphisms into a sheaf topos as claimed, the central claim holds; if the construction is restricted to cartesian or coherent theories, or if the proof fails for the theory of rings (whose classifying topos should be the Zariski topos), the advertised generality is not delivered.

Watch

Extended reading notes

Core claim

A topos—a category equivalent to sheaves on a site—is at once a generalized space and a universe with internal logic. The book derives this from familiar material: groups appear through actions, spaces through sheaves, and the Yoneda lemma turns objects into functors. Grothendieck topologies replace open sets by covering sieves; an axiomatic characterization recognizes toposes without a chosen site. The syntactic category of a geometric first-order theory, with the syntactic topology, builds a classifying topos: models of the theory in any topos correspond exactly to morphisms into it. This organizing result makes geometric logic and topos geometry one subject.

Load-bearing premise

The whole book assumes that the later, not-yet-visible chapters will supply full correct proofs of the standard theorems (recognizing toposes by axioms, describing morphisms into sheaf toposes, and building classifying toposes for any geometric theory), and that the reader already knows basic algebra, topology, and category theory.

Editorial extensions

If this is right

  • A reader who follows the five chapters should be able to treat topological spaces, schemes, and smooth or étale sites uniformly as categories of sheaves, and to compute with coverings, descent, and base change in one language.
  • Every geometric first-order theory has a classifying topos, unique up to equivalence; models of the theory in any topos are the same as geometric morphisms into that classifying topos.
  • Because a topos admits many site presentations, any topos can classify many different geometric theories; theories with equivalent classifying toposes are Morita-equivalent, making semantic equivalence a topos-theoretic fact.
  • Subtoposes of a classifying topos correspond to theory quotients obtained by adding axioms, so logical provability can be rephrased as a problem about generating Grothendieck topologies by sieves.
  • Structures expressible by finite limits and arbitrary colimits—internal rings, modules, sheaves of modules—are transported by inverse images of topos morphisms, preparing a unified framework for cohomology.

Reading between the lines

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

  • If the announced later chapters match the plan, this book offers a self-contained route from introductory category theory to classifying toposes; a student could be assessed by deriving the classifying-topos theorem from the earlier chapters without outside references.
  • The 'toposes as bridges' technique announced for the future version would make Morita equivalence a practical transfer method: any equivalence of classifying toposes becomes a license to move theorems between different mathematical contexts.
  • The reversible-viewpoints principle suggests a testable heuristic: whenever a notion is expressible both as a site-theoretic construction and as a syntactic construction in geometric logic, the two expressions should be connected by a canonical topos equivalence. Readers can test this on ring theory, group actions, or covering theory.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript is a French-language introductory book on sites and Grothendieck toposes, aimed at readers with background in algebra, topology, and basic category theory. The advertised arc moves from groups, topological spaces, and categories (Chapter I), to Grothendieck topologies and sheaves on sites (Chapter II), to the definition and categorical properties of toposes and Giraud's theorem (Chapter III), to the geometry of toposes and Diaconescu's equivalence (Chapter IV), and finally to geometric logic and classifying toposes (Chapter V). The version under review contains Chapter I in full, together with a detailed table of contents and chapter summaries. Chapter I develops the categorical foundations: categories, functors, the Yoneda lemma, representable functors, adjoint functors, limits and colimits, and relative categories, with applications to affine schemes and sheaves on topological spaces. The core standard results shown in the text are presented clearly and correctly: the Yoneda lemma, the affine-scheme sheaf construction, and the basic limit/colimit theorems. However, the manuscript does not yet contain the later chapters on which the book's full promise depends, and one of the first worked examples of adjoint functors contains a concrete error in the direction of the adjunction.

Significance. If completed and corrected, this would be a useful pedagogical contribution: the planned progression from elementary examples to classifying toposes is coherent, and the fully written Chapter I covers standard categorical material with care. The treatment of the Yoneda lemma and the affine-scheme sheaf lemma is competently done, and the many worked examples are a strength. The significance of the whole project rests on the promised later chapters, which are not present in this version. The adjunction error in §I.8.b is particularly damaging for a textbook whose explicit goal is to give readers a 'workable language' of adjunctions and universal properties, because a reader who studies this example will internalize the wrong direction of the adjunction. The error is local and fixable, but it must be corrected before the manuscript can serve its pedagogical purpose.

major comments (3)
  1. [§I.8.b, pp. 72–74] The subsection 'Le foncteur des points fixes et celui des orbites' states that the trivial-action functor T: Ens → BG (X ↦ X with trivial G-action) 'admet pour adjoint à gauche le foncteur des points fixes X^G et pour adjoint à droite le foncteur des orbites G\X'. This is backwards. The correct adjunctions are: Hom_BG(TX, Y) ≅ Hom_Ens(X, Y^G), since a G-equivariant map from a trivial G-set to Y factors through the fixed points, and Hom_BG(Y, TX) ≅ Hom_Ens(G\Y, X), since a G-equivariant map to a trivial G-set is constant on orbits. Thus (-)^G is the right adjoint of T and G\(-) is the left adjoint. This is not a harmless terminological slip: the example is explicitly presented as the first instance of a functor with distinct left and right adjoints, and the direction of the universal property is exactly what the reader is meant to learn. The correction should be made and any downstream re
  2. [Abstract and Introduction] The book's central promise, as stated in the abstract and the Introduction, is a progression 'pour aboutir aux fondements avancés de la géométrie des topos et à ses liens profonds avec la logique géométrique ... autour du théorème de construction des topos classifiants.' The version under review, however, contains only Chapter I in full. Chapters II–V are represented by a table of contents and synopses, not by the actual mathematical exposition. The claims about Giraud's theorem, Diaconescu's equivalence, and the classifying-topos theorem are therefore not supported by the text as submitted. If this is intended as a book proposal or first chapter, the title and abstract should state that explicitly; if it is intended as a complete introduction, the missing chapters are a substantive gap, not a presentational issue.
  3. [Chapter V summary] The summary of Chapter V says that the classifying-topos theorem 'repose sur l'équivalence de Diaconescu' and that parts follow 'le livre [TST]'. Since [TST] is a book by one of the present authors, the reader cannot check these central results from the manuscript itself. This may be acceptable in a full book that cites its own earlier work, but in the present standalone text—and in the absence of the full chapter—it means the advertised conclusion rests on an unstated external source. The authors should either supply the complete proofs or clearly frame the text as an excerpt that relies on [TST].
minor comments (4)
  1. [Chapter V summary] Typo: 'équivalence de ca catégorie' should read 'équivalence de la catégorie'.
  2. [Définition I.4.5] Typo: 'une space topologique' should be 'un espace topologique'.
  3. [General] The bibliography is missing: references such as [SGA 4], [TST], and [Categorical Logic] are cited by abbreviations but no reference list is included in the excerpt. A complete book manuscript should provide full bibliographic entries.
  4. [§I.8.b] The same subsection says 'On verra plus loin que pour tout foncteur ρ : C → D ... le foncteur ρ* admet un adjoint à droite ρ_* et un adjoint à gauche ρ_!'. This is announced as a later result, which is fine, but it might be helpful to flag explicitly that the fixed-point/orbit example is the group-action special case of the same pattern, once the direction is corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the book is an expository introduction whose load-bearing theorems are independently established outside the paper; the only self-citations are for exposition and are not load-bearing.

full rationale

The manuscript under review is an introduction (Chapter I plus a detailed plan). It does not derive new results from fitted parameters or rename empirical patterns. The arc from sites to toposes to classifying toposes rests on standard external theorems: Giraud's characterization, Diaconescu's equivalence, and the Makkai–Reyes existence theorem for classifying toposes, each explicitly attributed to established sources (SGA 4, the Montreal categorical-logic school, etc.). The few 'Suivant le livre [TST]' passages in Chapter V are self-citations to the first author's monograph, but they are expository shortcuts for standard results (subtoposes ↔ quotient theories, presheaf-type theories) whose proofs are available in the independently published literature and do not feed a parameter back into a prediction. There is no definitional equivalence between claimed outputs and inputs, no fitted quantity relabeled as a prediction, and no imported uniqueness theorem that forbids alternatives solely by authorial fiat. The noted §I.8.b swap of left/right adjoints for the trivial-action functor is a mathematical correctness concern, not a circularity; even if the pedagogical promise were undermined, the derivation chain does not reduce to its own assumptions. The stated plan to present 'topos as bridges' only in a future version is an explicit limitation, not a circular step.

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

The manuscript is an exposition, so the ledger records the unproved background theorems it rests on rather than fitted parameters. No free parameters and no invented entities appear. The book's announced theorems (Giraud, Diaconescu, classifying-topos existence) are standard results from the literature, stated in the excerpt but proved outside the reviewed portion. The pedagogical premise about the reader's background is the only non-mathematical assumption.

assumptions (5)
  • standard math A topos may be defined either as a category equivalent to sheaves on a site or, by Giraud's theorem, by an axiomatic list: locally small, finite limits, arbitrary colimits, effective equivalence relations, colimits stable under base change, and a separating family.
    The equivalence of the two definitions is the backbone of Chapter III (sections III.1 and III.8). The statement is standard from SGA 4; the proof is outside the reviewed portion.
  • standard math Diaconescu's equivalence: for any site, geometric morphisms from a topos into the sheaf topos of the site correspond to flat and continuous functors from the site's underlying category into the topos.
    Invoked in Chapter IV.3 and used in Chapter V.7 to build classifying toposes and to describe points of sheaf toposes. Standard categorical-logic theorem; proof not in the reviewed portion.
  • standard math Every first-order geometric theory has a classifying topos: its syntactic category equipped with the syntactic topology produces a topos whose model functor is representable.
    Central claim of Chapter V; attributed to Makkai and Reyes and to Monique Hakim's thesis. The excerpt states the theorem and says the proof rests on Diaconescu's equivalence; the proof itself is not visible in the reviewed portion.
  • standard math Grothendieck's comparison lemma: replacing a site by a full dense subcategory with the induced topology does not change the associated topos of sheaves.
    Cited in Chapter III.1 as a key technical principle; proof not in the reviewed portion.
  • domain assumption The intended reader has a reasonable familiarity with algebra (groups, rings, modules), general topology, and the rudiments of category theory.
    Stated in the section 'Public visé et fil méthodologique'; the book's didactic effectiveness depends on this prerequisite level.

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Cite this review

Pith. "Pith review of Sites and Grothendieck toposes: an introduction." pith.science (2026). https://pith.science/paper/VDULOQ4M

@misc{pith2026250821609,
  author       = {Pith},
  title        = {Pith review of: Sites and Grothendieck toposes: an introduction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VDULOQ4M}},
  note         = {Machine review of arXiv:2508.21609}
}
read the original abstract

Topos theory occupies a singular place in contemporary mathematics: born from Grothendieck's algebraic geometry, it has emerged as a unifying language for geometry, topology, algebra, and logic. This book offers a progressive introduction that moves from familiar ground - groups and their actions, topological spaces, categories, and sheaves - to Grothendieck topologies and sites, then to the axiomatic and categorical foundations of toposes (via Giraud's theorem), to their geometry (morphisms, points, subtoposes, localizations) and finally to their deep ties with geometric logic through classifying toposes. Aimed at readers with a basic familiarity with algebra, general topology, and category theory, the book emphasizes reversible viewpoints - external/internal, local/global, syntactic/semantic - guided by canonical examples, key theorems, and universal constructions. It equips the reader with a workable language in which spaces become categories of sheaves and theories become places classified by toposes. Future chapters will present "toposes as bridges" and relative toposes.

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Reference graph

Works this paper leans on

22 extracted references · 21 canonical work pages

  1. [2]

    – Grothendieck Soient T1 et T2 deux th´ eories alg´ ebriques [resp

    CAT ´EGORIES SYNTACTIQUES 569 Grothendieck On d´ eduit des propositions V.6.16 et V.6.17 : Corollaire V.6.19. – Grothendieck Soient T1 et T2 deux th´ eories alg´ ebriques [resp. cart´ esiennes, resp. r´ eguli` eres, resp. coh´ erentes, resp. g´ eom´ etriques, resp. du premier ordre finitaires] dans des signaturesΣ1 et Σ2. Alors ces th´ eories sont syntact...

    work page

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    – Grothendieck Soit C une cat´ egorie r´ eguli` ere [resp

    TOPOS CLASSIFIANTS 573 Grothendieck V´ erifions que la d´ efinition V.7.2 ci-dessus est valide : Proposition V.7.3. – Grothendieck Soit C une cat´ egorie r´ eguli` ere [resp. coh´ erente, resp. g´ eom´ etrique] essentiellement petite. Alors : (i) Il existe une topologie de Grothendieck J de C pour laquelle un crible sur un objet X est couvrant s’il contie...

    work page

  3. [5]

    Ces deux applications sont inverses l’une de l’autre

    TOPOS CLASSIFIANTS 575 Dans l’autre sens, toute famille de sous-objets X ′ i ,− →Xi qui co ¨ ıncident dans lesXi ×S Xj d´ efinit un sous-objet X ′ ,− →X qui est la r´ eunion des images desX ′ i ,→ Xi par les morphismes Xi → X. Ces deux applications sont inverses l’une de l’autre. Si X ′ ,→ X est un sous-objet, le fait que X soit r´ eunion des images desXi...

    work page

  4. [6]

    Dans l’autre sens, partons d’un foncteur cart´ esien [resp

    CAT ´EGORIES SYNTACTIQUES 567 Il en r´ esulte que le compos´ e M 7− →FM (MT) du foncteur M 7→ FM suivi du foncteur F 7→ F (MT) n’est autre que le foncteur d’identit´ e de la cat´ egorie T-mod (C) M 7− →M . Dans l’autre sens, partons d’un foncteur cart´ esien [resp. r´ egulier, resp. coh´ erent, resp. g´ eom´ etrique, resp. de Heyting] F : CT − → C. Le mod...

    work page

  5. [7]

    Elle se d´ emontre comme la partie d’unicit´ e du th´ eor` eme V.6.1

    TOPOS CLASSIFIANTS 571 Commencement de la d´ emonstration: : Grothendieck Prouvons d’abord l’unicit´ e ` a ´ equivalence pr` es des toposET v´ erifiant la propri´ et´ e de l’´ enonc´ e. Elle se d´ emontre comme la partie d’unicit´ e du th´ eor` eme V.6.1. Si ET et E ′ T sont deux topos munis de mod` elesUT et U ′ T de T qui satisfont chacun la propri´ et´...

    work page

  6. [8]

    coh´ erente, resp

    TOPOS CLASSIFIANTS 577 Grˆ ace ` a ce lemme, on d´ eduit de la proposition V.6.17 et de l’´ equivalence de Diaconescu que le topos ET associ´ e au site constitu´ e de la cat´ egorie syntactiqueCT d’une th´ eorie g´ eom´ etrique [resp. coh´ erente, resp. r´ eguli` ere, resp. cart´ esienne]T et de sa topologie syntactique JT v´ erifie la propri´ et´ e de re...

    work page

  7. [9]

    Or, d’apr` es le corollaire V.7.4, le foncteur canonique ℓ : CT − → ET est pleinement fid` ele

    TOPOS CLASSIFIANTS 579 ment si, dans la cat´ egorieCT, les deux sous-objets MT φ(⃗ x) et MT ψ(⃗ x) de l’objet MT⊤(⃗ x) satisfont la relation d’inclusion MT φ(⃗ x) ≤ MT ψ(⃗ x) . Or, d’apr` es le corollaire V.7.4, le foncteur canonique ℓ : CT − → ET est pleinement fid` ele. De plus, il respecte les limites finies. Il en r´ esulte que deux sous-objets d’un o...

    work page

  8. [10]

    Cela r´ esulte du th´ eor` eme suivant : Th´ eor` eme V.7.10

    TOPOS CLASSIFIANTS 581 f ) Pr´ esentation des topos par des th´ eories R´ eciproquement, tout topos s’´ ecrit comme le topos classifiant d’une infinit´ e de th´ eories g´ eom´ etriques du premier ordre. Cela r´ esulte du th´ eor` eme suivant : Th´ eor` eme V.7.10. –Grothendieck Soit C une petite cat´ egorie munie d’une topologie de GrothendieckJ. Soit ΣC ...

    work page

Show all 22 references
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    LOGIQUE G ´EOM ´ETRIQUE ET TOPOLOGIES DE GROTHENDIECK 583 (iii) Un foncteur ` a valeurs dans un topos E M : C − → E est J-continu s’il transforme toute famille J-couvrante de morphismes de C fi : Xi − →X , i ∈ I , en une famille globalement ´ epimorphique. Cela revient ` a dem...

  2. [12]

    LOGIQUE G ´EOM ´ETRIQUE ET TOPOLOGIES DE GROTHENDIECK 585 Associons ` a toute th´ eorie quotientT′ de T la topologie sur CT JT′ ⊇ JT engendr´ ee parJT et par les recouvrements (φ ∧ ψ)(⃗ x) ,− − →φ(⃗ x) associ´ es aux axiomes φ ⊢⃗ xψ de T′. En sens inverse, associons ` a toute ...

  3. [13]

    Cela ach` eve la preuve de (i)

    LOGIQUE G ´EOM ´ETRIQUE ET TOPOLOGIES DE GROTHENDIECK 587 Comme il en est de mˆ eme du s´ equent φ ⊢⃗ x _ i∈I (∃ ⃗ xi) θi(⃗ xi, ⃗ x) , on obtient que le s´ equent φ ⊢⃗ x _ i∈I _ j∈I ′ (∃ ⃗ xi)(∃ ⃗ yj)(θ′ j(⃗ yj, ⃗ x) ∧ θi(⃗ xi, ⃗ x)) est T′-d´ emontrable, donc a fortiori le s´...

  4. [14]

    – Grothendieck Soit T une th´ eorie g´ eom´ etrique de signatureΣ

    LOGIQUE G ´EOM ´ETRIQUE ET TOPOLOGIES DE GROTHENDIECK 589 b) D´ emontrabilit´ e et topologies engendr´ ees Le th´ eor` eme V.8.1 permet d’exprimer les probl` emes de d´ emontrabilit´ e dans les th´ eories g´ eom´ etriques comme des probl` emes de topologies de Grothendieck : C...

  5. [15]

    th´ eories de type pr´ efaisceau

    TH ´EORIES DE TYPE PR ´EF AISCEAU 591 9 Th´ eories de type pr´ efaisceau a) La notion de th´ eorie de type pr´ efaisceau Toute th´ eorie g´ eom´ etrique du premier ordreT admet un topos classifiant ET. Pour n’importe quelle pr´ esentation de celui-ci comme topos des faisceaux ...

  6. [16]

    finiment pr´ esentable

    TH ´EORIES DE TYPE PR ´EF AISCEAU 593 En effet, toute ´ equivalence de topos ET ∼= bC induit en particulier par d´ efinition des topos classifiants une ´ equivalence de cat´ egories T-mod (Ens) ∼= pt( bC) . Or, d’apr` es le th´ eor` eme IV.3.8, la cat´ egorie des points du top...

  7. [17]

    compl´ etion Karoubienne

    TH ´EORIES DE TYPE PR ´EF AISCEAU 595 (i) Pour toute telle cat´ egorie essentiellement petiteC, la sous-cat´ egorie pleine de Ind (C) ou bC constitu´ ee des r´ etractes d’objets repr´ esentables est appel´ ee la “compl´ etion Karoubienne” deC et peut ˆ etre not´ ee Kar (C) . (...

  8. [18]

    irr´ eductible

    TH ´EORIES DE TYPE PR ´EF AISCEAU 597 (i) L’´ equivalence de cat´ egories induite T-mod(Ens) ∼ − − →Ind(Cop) se restreint en une ´ equivalence T-mod(Ens)fp ∼ − − →Kar(Cop) de la cat´ egorie des mod` eles ensemblistes finiment pr´ esentables deT sur la compl´ etion Karoubienne ...

  9. [19]

    Grothendieck Cette d´ efinition ´ etant pos´ ee, on peut ´ enoncer le crit` ere suivant : Th´ eor` eme V.9.11

    TH ´EORIES DE TYPE PR ´EF AISCEAU 599 (iii) Par d´ efinition de la structure cat´ egorique deCT et de sa topologie g´ eom´ etriqueJT, une formule g´ eom´ etrique de la signature Σ de T φ(⃗ x) est T-irr´ eductible si et seulement si, pour toute famille de formules d´ emontrable...

  10. [20]

    finiment pr´ esent´ e

    TH ´EORIES DE TYPE PR ´EF AISCEAU 601 A fortiori, la sous-cat´ egorie pleineCir T de CT est JT-dense, comme on voulait. □ On d´ eduit de ce th´ eor` eme et du lemme : Corollaire V.9.13. – Grothendieck Soit T une th´ eorie g´ eom´ etrique de type pr´ efaisceau. Soient CT = Cgeo...

  11. [21]

    Via l’´ equivalenceET ∼ − − →[Mop, il s’identifie au point du topos classifiant Ens − → ET qui correspond ` aM vu comme un mod` ele ensembliste deT

    TH ´EORIES DE TYPE PR ´EF AISCEAU 603 D´ emonstration du lemme : Grothendieck (i) Tout objet M de Mop d´ efinit un point du topos[Mop c’est-` a-dire un morphisme de topos Ens − →[Mop . Via l’´ equivalenceET ∼ − − →[Mop, il s’identifie au point du topos classifiant Ens − → ET q...

  12. [22]

    TH ´EORIES DE TYPE PR ´EF AISCEAU 605 (iv) R´ eciproquement, pour toute th´ eorie g´ eom´ etriqueT de signature Σ, toute formule g´ eom´ etrique de Σ φ(⃗ x) de contexte ⃗ x= (xA1 1 · · ·xAn n ) d´ efinit une propri´ et´ e fonctorielle des mod` eles ensemblistes deT M 7− →(M φ(...

  13. [23]

    R´ eciproquement, consid´ erons une th´ eorie g´ eom´ etriqueT qui satisfait les conditions (1), (2) et (3)

    TH ´EORIES DE TYPE PR ´EF AISCEAU 607 Cela ach` eve de prouver (3). R´ eciproquement, consid´ erons une th´ eorie g´ eom´ etriqueT qui satisfait les conditions (1), (2) et (3). Notons (CT, JT) le site syntactique g´ eom´ etrique deT et M = T-mod(Ens)fp la cat´ egorie des mod` ...

  14. [24]

    Th´ eorie des topos

    TH ´EORIES DE TYPE PR ´EF AISCEAU 609 D’apr` es la condition (1), cela implique que le s´ equent φ ⊢⃗ x _ i∈I (∃ ⃗ xi) θi(⃗ xi, ⃗ x) est T-d´ emontrable. Autrement dit, la famille des morphismes de CT θi(⃗ xi, ⃗ x) : φi(⃗ xi) − →φ(⃗ x) est JT-couvrante, et l’objet φ(⃗ x) admet...

    1973 arXiv

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