pith. sign in

arxiv: 2507.01869 · v2 · submitted 2025-07-02 · 🧮 math.CT · math.LO

De Morgan's law in toposes I

Olivia Caramello , Yorgo Chamoun This is my paper

Pith reviewed 2026-05-19 06:32 UTC · model grok-4.3

classification 🧮 math.CT math.LO
keywords De Morgan toposgeometric theoryclassifying toposamalgamation propertycategory of modelstopos theorycategorical logic
0
0 comments X

The pith

Geometric theories have De Morgan classifying toposes precisely when their model categories satisfy amalgamation.

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

The paper gives characterizations of geometric theories whose classifying topos satisfies De Morgan's law. It clarifies the connection between this property and the amalgamation property of the category of models of the theory. A reader would care because the result ties a logical feature of the topos directly to a model-theoretic condition on the theory, which can help identify when certain logical simplifications occur in categorical logic. The work also supplies several explicit constructions that turn an arbitrary topos into a De Morgan topos.

Core claim

We give characterizations of geometric theories whose classifying topos is De Morgan, clarifying the link with the amalgamation property of the category of models of such theory. We then give several ways of turning a topos into a De Morgan topos.

What carries the argument

The classifying topos of a geometric theory, whose satisfaction of De Morgan's law is shown to correspond to the amalgamation property in the category of models.

If this is right

  • Geometric theories can now be checked for a De Morgan classifying topos by verifying amalgamation in their model categories.
  • The internal logic of the classifying topos obeys De Morgan's law exactly when amalgamation holds for the models.
  • Explicit constructions exist that convert any topos into one satisfying De Morgan's law.

Where Pith is reading between the lines

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

  • The same link may allow similar characterizations for other logical principles inside classifying toposes.
  • The result could be used to locate new families of geometric theories whose models admit amalgamation.
  • It suggests examining whether amalgamation controls additional features of the internal logic beyond De Morgan's law.

Load-bearing premise

The De Morgan property of the classifying topos can be related directly to amalgamation in the model category without further restrictions on the theory or the topos.

What would settle it

A geometric theory whose category of models has the amalgamation property yet whose classifying topos fails to satisfy De Morgan's law, or the converse situation.

read the original abstract

We study toposes satisfying De Morgan's law, in particular we give characterizations of geometric theories whose classifying topos is De Morgan, clarifying the link with the amalgamation property of the category of models of such theory. We then give several ways of turning a topos into a De Morgan topos.

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 / 0 minor

Summary. The paper studies toposes satisfying De Morgan's law. It gives characterizations of geometric theories whose classifying topos is De Morgan and clarifies the connection to the amalgamation property of the category of models of the theory. It also presents several constructions for converting an arbitrary topos into a De Morgan topos.

Significance. If the characterizations hold, the work would provide a useful bridge between the logical property of De Morgan toposes and model-theoretic amalgamation, with potential applications in the study of classifying toposes and geometric logic. The explicit constructions for producing De Morgan toposes add constructive value.

major comments (1)
  1. [§§3–4] §§3–4: The claimed equivalence between the De Morgan property of the classifying topos and the amalgamation property in the category of models is stated for general geometric theories, but the argument appears to rely on the syntactic category being finitely complete or the theory being coherent. For infinitary geometric theories, infinitary disjunctions can produce models where filtered colimits fail to preserve the relevant pushouts, so amalgamation on finitely presented models need not lift; an explicit restriction or counterexample handling is needed to support the general claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and insightful comments on our manuscript. The major comment raises an important point about the scope of our characterization, which we address below.

read point-by-point responses

  1. Referee: [§§3–4] §§3–4: The claimed equivalence between the De Morgan property of the classifying topos and the amalgamation property in the category of models is stated for general geometric theories, but the argument appears to rely on the syntactic category being finitely complete or the theory being coherent. For infinitary geometric theories, infinitary disjunctions can produce models where filtered colimits fail to preserve the relevant pushouts, so amalgamation on finitely presented models need not lift; an explicit restriction or counterexample handling is needed to support the general claim.

    Authors: We appreciate the referee highlighting this subtlety. Our argument in §§3–4 does rely on the syntactic category being finitely complete (hence on the theory being coherent), as the relevant pushouts and filtered colimits are preserved in that setting. For fully infinitary geometric theories the referee's concern is valid: infinitary disjunctions can yield models in which amalgamation on finitely presented objects fails to lift to the De Morgan property of the classifying topos. We therefore agree that the general claim as stated requires qualification. In the revised version we will explicitly restrict the equivalence to coherent geometric theories, add a remark explaining why the infinitary case needs additional hypotheses or separate treatment, and include a brief discussion of the obstruction the referee describes. This preserves the main results while accurately delimiting their scope. revision: yes

Circularity Check

0 steps flagged

No significant circularity; characterizations derived independently from topos and model-category axioms

full rationale

The paper supplies explicit characterizations of geometric theories whose classifying toposes satisfy De Morgan's law and establishes their connection to the amalgamation property via syntactic-site constructions and filtered-colimit arguments in the category of models. These steps rely on standard definitions of geometric theories, classifying toposes, and the De Morgan condition (¬(A ∧ B) = ¬A ∨ ¬B on the subobject classifier) together with the usual properties of finitely presented models and pushouts; none of the central equivalences or links are obtained by re-labeling fitted parameters, by self-referential definitions, or by load-bearing self-citations whose content is presupposed. The derivation therefore remains self-contained against external benchmarks in topos theory and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract mentions no free parameters, specific axioms beyond standard topos theory, or invented entities.

pith-pipeline@v0.9.0 · 5557 in / 1003 out tokens · 70912 ms · 2026-05-19T06:32:12.609498+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · 1 internal anchor

  1. [1]

    Locally presentable and accessible categories , volume 189

    J Adamek and J Rosicky. Locally presentable and accessible categories , volume 189. Cambridge University Press, 1994

    work page 1994

  2. [2]

    De M organ's law and related identities in classifying topoi

    A Bagchi. De M organ's law and related identities in classifying topoi. In Canadian Mathematical Society Conf. Proceeding , volume 13, pages 1--31, 1992

    work page 1992

  3. [3]

    Classifying toposes for first-order theories

    Carsten Butz and Peter Johnstone. Classifying toposes for first-order theories. Annals of Pure and Applied Logic , 91(1):33--58, 1998

    work page 1998

  4. [4]

    Handbook of Categorical Algebra: Volume 3, Sheaf Theory , volume 3

    Francis Borceux. Handbook of Categorical Algebra: Volume 3, Sheaf Theory , volume 3. Cambridge university press, 1994

    work page 1994

  5. [5]

    Categories of set valued functors

    Marta Cavallo Bunge. Categories of set valued functors . University of Pennsylvania, 1966

    work page 1966

  6. [6]

    De M organ classifying toposes

    Olivia Caramello. De M organ classifying toposes. Advances in Mathematics , 222(6):2117--2144, 2009

    work page 2009

  7. [7]

    A topos-theoretic approach to Stone-type dualities

    Olivia Caramello. A topos-theoretic approach to S tone-type dualities. arXiv preprint arXiv:1103.3493 , 2011

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  8. [8]

    Syntactic characterizations of properties of classifying toposes

    Olivia Caramello. Syntactic characterizations of properties of classifying toposes. Theory & Applications of Categories , 26(2), 2012

    work page 2012

  9. [9]

    Theories, S ites, T oposes: R elating and studying mathematical theories through topos-theoretic'bridges'

    Olivia Caramello. Theories, S ites, T oposes: R elating and studying mathematical theories through topos-theoretic'bridges' . Oxford University Press, 2018

    work page 2018

  10. [10]

    Denseness conditions, morphisms and equivalences of toposes

    Olivia Caramello. Denseness conditions, morphisms and equivalences of toposes. arXiv preprint arXiv:1906.08737 , 2019

    work page arXiv 1906

  11. [11]

    Fibred sites and existential toposes

    Olivia Caramello. Fibred sites and existential toposes. arXiv preprint arXiv:2212.11693 , 2022

    work page arXiv 2022

  12. [12]

    De M organ's law and the theory of fields

    Olivia Caramello and Peter Johnstone. De M organ's law and the theory of fields. Advances in Mathematics , 222(6):2145--2152, 2009

    work page 2009

  13. [13]

    Relative topos theory via stacks

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

    work page arXiv 2021

  14. [14]

    Every theory is eventually of presheaf type

    Christian Esp \' ndola and Krist \'o f Kanalas. Every theory is eventually of presheaf type. arXiv preprint arXiv:2312.12356 , 2023

    work page arXiv 2023

  15. [15]

    Infinitary generalizations of D eligne’s completeness theorem

    Christian Esp \' ndola. Infinitary generalizations of D eligne’s completeness theorem. The Journal of Symbolic Logic , 85(3):1147--1162, 2020

    work page 2020

  16. [16]

    Classifying T opos

    Jean Giraud. Classifying T opos. Toposes, Algebraic Geometry and Log ic , page 43, 1972

    work page 1972

  17. [17]

    Sga 4 expose iv

    A Grothendieck and JL Verdier. Sga 4 expose iv. Theorie des Topos et Cohomologie Etale des Schemas. Seminaire de Geometrie Algebrique du Bois-Marie 1963-1964 (SGA 4): Tome 1 , 269:1, 2006

    work page 1963

  18. [18]

    Applications of D e M organ toposes and the G leason cover

    Rona Harun. Applications of D e M organ toposes and the G leason cover. 1996

    work page 1996

  19. [19]

    The G leason cover of a topos, I

    Peter T Johnstone. The G leason cover of a topos, I . Journal of Pure and Applied Algebra , 19:171--192, 1980

    work page 1980

  20. [20]

    Sketches of an E lephant: A T opos T heory Compendium: Volume 2 , volume 2

    Peter T Johnstone. Sketches of an E lephant: A T opos T heory Compendium: Volume 2 , volume 2. Oxford University Press, 2002

    work page 2002

  21. [21]

    Conditions related to D e M organ's law

    Peter T Johnstone. Conditions related to D e M organ's law. In Applications of Sheaves: Proceedings of the Research Symposium on Applications of Sheaf Theory to Logic, Algebra, and Analysis, Durham, July 9--21, 1977 , pages 479--491. Springer, 2006

    work page 1977

  22. [22]

    Existentially closed models and locally zero-dimensional toposes

    Mark Kamsma and Joshua Wrigley. Existentially closed models and locally zero-dimensional toposes. arXiv preprint arXiv:2406.02788 , 2024

    work page arXiv 2024

  23. [23]

    Sheaves in geometry and logic: A first introduction to topos theory

    Saunders MacLane and Ieke Moerdijk. Sheaves in geometry and logic: A first introduction to topos theory . Springer Science & Business Media, 2012

    work page 2012

  24. [24]

    First order categorical logic: model-theoretical methods in the theory of topoi and related categories , volume 611

    Michael Makkai and Gonzalo E Reyes. First order categorical logic: model-theoretical methods in the theory of topoi and related categories , volume 611. Springer, 2006

    work page 2006

  25. [25]

    The model-theoretic significance of complemented existential formulas

    Volker Weispfenning. The model-theoretic significance of complemented existential formulas. The Journal of Symbolic Logic , 46(4):843--850, 1981

    work page 1981

  26. [26]

    Some properties of internal locale morphisms externalised

    Joshua Wrigley. Some properties of internal locale morphisms externalised. arXiv preprint arXiv:2301.00961 , 2023

    work page arXiv 2023

  27. [27]

    Fondements de la logique positive

    Ita \" Ben Yaacov and Et Bruno Poizat. Fondements de la logique positive. The Journal of Symbolic Logic , 72(4):1141--1162, 2007

    work page 2007