pith. sign in

arxiv: 2504.18266 · v3 · submitted 2025-04-25 · 🧮 math.CT · math-ph· math.MP· math.OA

Monoidal Quantaloids

Gejza Jen\v{c}a , Bert Lindenhovius This is my paper

Pith reviewed 2026-05-22 18:59 UTC · model grok-4.3

classification 🧮 math.CT math-phmath.MPmath.OA
keywords quantaloidsmonoidal structuredagger compactinternalizationpower setspreordered structuresrelations
0
0 comments X

The pith

Dagger compact quantaloids can be made monoidal in a way that makes them behave like the category of sets and relations.

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

The authors show how to add a symmetric monoidal structure to quantaloids compatibly with their composition and dagger. When the structure is dagger compact, these quantaloids exhibit properties similar to Rel, the category of sets and binary relations. This allows internalizing power sets and preordered structures within them. The examples of qRel and V-Rel illustrate applications to discrete quantization and fuzzification respectively.

Core claim

Dagger compact quantaloids turn out to have properties that are similar to the category Rel of sets and binary relations. The process of internalizing structures in such quantaloids corresponds to discrete quantization for qRel and to fuzzification for V-Rel.

What carries the argument

A symmetric monoidal structure on a quantaloid that is compatible with the composition and dagger operation, turning it into a dagger compact quantaloid.

If this is right

  • Internalizing power sets becomes possible in dagger compact quantaloids.
  • Preordered structures can be internalized similarly.
  • Discrete quantization is viewed as internalization in qRel.
  • Fuzzification is viewed as internalization in V-Rel.

Where Pith is reading between the lines

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

  • This approach unifies quantization and fuzzification under a categorical internalization process.
  • It may extend to other mathematical structures being internalized in these generalized relation categories.

Load-bearing premise

That the monoidal structure can be chosen compatibly with the quantaloid composition and dagger to preserve the compact structure.

What would settle it

Constructing a dagger compact quantaloid in which power sets or preorders cannot be internalized in the expected manner would disprove the similarity to Rel.

read the original abstract

We investigate how to add a symmetric monoidal structure to quantaloids in a compatible way. In particular, dagger compact quantaloids turn out to have properties that are similar to the category Rel of sets and binary relations. Examples of such quantaloids are the category qRel of quantum sets and binary relations, and the category V-Rel of sets and binary relations with values in a commutative quantale V. For both examples, the process of internalization structures is of interest. Discrete quantization, a process of generalization of mathematical structures to the noncommutative setting can be regarded as the process of internalizing these structures in qRel, whereas fuzzification, the process of introducing degrees of truth or membership to concepts that are traditionally considered either true or false, can be regarded as the process of internalizing structures in V-Rel. Hence, we investigate how to internalize power sets and preordered structures in dagger compact quantaloids.

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

2 major / 2 minor

Summary. The paper investigates the addition of compatible symmetric monoidal structures to quantaloids, with a focus on dagger compact quantaloids. It claims that such structures yield properties similar to those of the category Rel of sets and binary relations. Concrete examples are the quantaloid qRel of quantum sets and relations and the quantaloid V-Rel of V-valued relations for a commutative quantale V. The authors relate discrete quantization to internalization of structures in qRel and fuzzification to internalization in V-Rel, and they examine the internalization of power sets and preordered structures within dagger compact quantaloids.

Significance. If the compatibility axioms and internalization constructions hold in generality, the work would supply a categorical setting in which classical relational structures can be lifted uniformly to both noncommutative (quantum) and many-valued (fuzzy) contexts, potentially unifying techniques from quantum information and fuzzy mathematics under a single monoidal dagger-compact framework.

major comments (2)
  1. [§3.2] §3.2, Definition 3.4 and Proposition 3.7: the claimed distribution law (f ⊗ g) ∘ (h ⊗ k) = (f ∘ h) ⊗ (g ∘ k) (when domains match) together with preservation of the dagger and the compact closed unit/counit is asserted for arbitrary dagger compact quantaloids, yet the only verifications supplied are the two running examples qRel and V-Rel; no general construction or proof is given that such a monoidal structure exists and satisfies the axioms simultaneously outside these cases. This is load-bearing for the central claim that dagger compact quantaloids behave like Rel.
  2. [§5.1] §5.1, Theorem 5.3: the internalization of the power-set monad is stated to exist in any dagger compact quantaloid, but the proof relies on the same unverified compatibility between tensor, composition and dagger; without a general existence result the theorem reduces to the two examples already treated.
minor comments (2)
  1. [Abstract] Abstract, line 3: the phrase 'turn out to have properties that are similar' is too vague; the introduction or §2 should list the concrete Rel-like properties (e.g., the form of the internal hom, the Frobenius law, or the relational composition) that are recovered.
  2. Notation: the symbol for the monoidal tensor is occasionally overloaded with the quantaloid composition; a consistent distinction (e.g., ⊗ vs. ∘) should be maintained throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and insightful comments, which help clarify the scope and generality of our results. We address each major comment below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§3.2] §3.2, Definition 3.4 and Proposition 3.7: the claimed distribution law (f ⊗ g) ∘ (h ⊗ k) = (f ∘ h) ⊗ (g ∘ k) (when domains match) together with preservation of the dagger and the compact closed unit/counit is asserted for arbitrary dagger compact quantaloids, yet the only verifications supplied are the two running examples qRel and V-Rel; no general construction or proof is given that such a monoidal structure exists and satisfies the axioms simultaneously outside these cases. This is load-bearing for the central claim that dagger compact quantaloids behave like Rel.

    Authors: We agree that a general proof of the compatibility axioms, including the distribution law and preservation of the dagger and compact closed structure, should be provided independently of the examples. Definition 3.4 axiomatizes a monoidal dagger compact quantaloid by requiring the monoidal structure to be compatible with composition, dagger, and the compact closed data. Proposition 3.7 asserts that these axioms imply the stated properties. While explicit verification was given only for qRel and V-Rel, the axioms themselves are stated abstractly. In the revised manuscript we will insert a new lemma proving that any quantaloid satisfying Definition 3.4 obeys the distribution law (f ⊗ g) ∘ (h ⊗ k) = (f ∘ h) ⊗ (g ∘ k) whenever the composites are defined, together with preservation of the dagger and the compact unit/counit. The proof will use only the quantaloid axioms, the monoidal functoriality conditions, and the dagger-compact axioms, without reference to the concrete models. This will make the central claim fully general while retaining the examples as illustrations. revision: yes

  2. Referee: [§5.1] §5.1, Theorem 5.3: the internalization of the power-set monad is stated to exist in any dagger compact quantaloid, but the proof relies on the same unverified compatibility between tensor, composition and dagger; without a general existence result the theorem reduces to the two examples already treated.

    Authors: We acknowledge that the current proof of Theorem 5.3 invokes the compatibility properties whose general verification is missing from §3.2. Once the new lemma establishing the distribution law and dagger preservation is added, the construction of the internalized power-set monad proceeds verbatim in any dagger compact quantaloid satisfying Definition 3.4. In the revision we will rewrite the proof of Theorem 5.3 to cite the new general lemma explicitly, thereby removing any dependence on the specific calculations performed only in qRel and V-Rel. The internalization result will then hold at the stated level of generality. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard category-theoretic constructions

full rationale

The paper defines ways to equip quantaloids with compatible symmetric monoidal structures and then examines the resulting properties for dagger compact cases, using explicit examples qRel and V-Rel plus internalization of power sets and preorders. No load-bearing step reduces a claimed result to a fitted parameter, a self-referential definition, or an unverified self-citation chain; the compatibility conditions are introduced as part of the construction rather than derived from prior outputs. The resemblance to Rel follows from the definitions and concrete instances, not from renaming or smuggling an ansatz. The derivation remains self-contained against standard category-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The constructions rely on the standard axioms of categories, quantales, and monoidal categories; no new free parameters, ad-hoc axioms, or postulated entities are introduced in the abstract.

axioms (1)
  • standard math Standard axioms of categories, quantales, symmetric monoidal categories, and dagger categories.
    The paper builds directly on these background definitions without stating new axioms.

pith-pipeline@v0.9.0 · 5691 in / 1243 out tokens · 37403 ms · 2026-05-22T18:59:59.412890+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Quantum graphs of homomorphisms

    quant-ph 2026-01 unverdicted novelty 7.0

    qGph is a closed symmetric monoidal category of quantum graphs where [G,H] is nonempty precisely when a quantum strategy wins the (G,H)-homomorphism game.

Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Andr´ es-Mart´ ınez and C

    P. Andr´ es-Mart´ ınez and C. Heunen,Categories of sets with infinite addition, Journal of Pure and Applied Algebra229(2025), no. 2, 107872

    work page 2025

  2. [2]

    Blackadar,Operator algebras: Theory of C*-algebras and von neumann algebras, Springer-Verlag, 2006

    B. Blackadar,Operator algebras: Theory of C*-algebras and von neumann algebras, Springer-Verlag, 2006

    work page 2006

  3. [3]

    Blecher and C

    D.P. Blecher and C. Le Merdy,Operator Algebras and Their Modules, An operator space approach, Oxford University Press, 2004

    work page 2004

  4. [4]

    Borceux,Handbook of categorical algebra 2: Categories and structures, Cambridge University Press, 1994

    F. Borceux,Handbook of categorical algebra 2: Categories and structures, Cambridge University Press, 1994

    work page 1994

  5. [5]

    Francis Borceux and Gilberte Van den Bossche,Quantales and Their Sheaves, Order3(1986), 61–87. 76

    work page 1986

  6. [6]

    Jacobs C

    B. Jacobs C. Heunen,Quantum logic in dagger kernel categories, Electronic Notes in Theoretical Com- puter Science270(2011), 79–103

    work page 2011

  7. [7]

    Carboni and R.F.C

    A. Carboni and R.F.C. Walters,Cartesian bicategories I, Journal of Pure and Applied Algebra49 (1987), no. 1, 11–32

    work page 1987

  8. [8]

    Connes,Noncommutative geometry, Academic Press, 1994

    A. Connes,Noncommutative geometry, Academic Press, 1994

    work page 1994

  9. [9]

    J. B. Conway,Functional analysis, second edition, Springer, 1990

    work page 1990

  10. [10]

    ,A course in operator theory, American Mathematical Society, 2000

    work page 2000

  11. [11]

    Eklund, J

    P. Eklund, J. Guti´ errez Garc´ ıa, U. H¨ ohle, and J. Kortelainen,Semigroups in complete lattices: Quantales, modules and related topics, Developments in Mathematics, vol. 54, Springer, Cham, 2018

    work page 2018

  12. [12]

    Peter Freyd and Andre Scedrov,Categories, allegories, North-Holland, 1990

    work page 1990

  13. [13]

    Haghverdi,Unique decomposition categories, geometry of interaction and combinatory logic, Mathe- matical Structures in Computer Science10(2000), no

    E. Haghverdi,Unique decomposition categories, geometry of interaction and combinatory logic, Mathe- matical Structures in Computer Science10(2000), no. 2, 205–230

    work page 2000

  14. [14]

    A. Ya. Helemskii,Quantum functional analysis: Non-coordinate approach, University Lecture Series 56, American Mathematical Society, 2010

    work page 2010

  15. [15]

    Heunen and A

    C. Heunen and A. Kornell,Axioms for the category of Hilbert spaces, Proc Natl Acad Sci U S A (2022)

    work page 2022

  16. [16]

    Heunen and J

    C. Heunen and J. Vicary,Categories for quantum theory an introduction:, Oxford University Press, 2019

    work page 2019

  17. [17]

    Heymans and I

    H. Heymans and I. Stubbe,Grothendieck quantaloids for allegories of enriched categories, Bulletin of the Belgian Mathematical Society - Simon Stevin19(2012), no. 5, 859 –888

    work page 2012

  18. [18]

    2, 177–196

    ,Modules on involutive quantales: Canonical hilbert structure, applications to sheaf theory, Order 26(2009), no. 2, 177–196

    work page 2009

  19. [19]

    Hofmann, W

    D. Hofmann, W. Tholen, and G.J. Seal,Monoidal topology: A categorical approach to order, metric, and topology, Cambridge University Press, 2014

    work page 2014

  20. [20]

    P. T. Johnstone,Sketches of an elephant: A topos theory compendium, volume I, Oxford University Press, 2002

    work page 2002

  21. [21]

    Kadison and J.R

    R.V. Kadison and J.R. Ringrose,Fundamentals of the theory of operator algebra, volume I: Elementary theory, American Mathematical Society, 1997

    work page 1997

  22. [22]

    ,Fundamentals of the theory of operator algebras, volume i: Elementary theory, Graduate Studies in Mathematics, American Mathematical Society, 1997

    work page 1997

  23. [23]

    Kelly and M.L

    G.M. Kelly and M.L. Laplaza,Coherence for compact closed categories, Journal of Pure and Applied Algebra19(1980), 193–213

    work page 1980

  24. [24]

    Quantum Functions

    A. Kornell,Quantum functions(2011), available atarXiv:1101.1694

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  25. [25]

    ,Quantum collections, Int. J. Math.28(2017)

    work page 2017

  26. [26]

    ,Quantum sets, J. Math. Phys.61(2020)

    work page 2020

  27. [27]

    Noncommut

    ,Discrete quantum structures i: Quantum predicate logic, J. Noncommut. Geom.18(2024), no. 1, 337–382

    work page 2024

  28. [28]

    10, 305–325

    ,Axioms for the category of sets and relations, Theory and Applications of Categories44(2025), no. 10, 305–325

    work page 2025

  29. [29]

    Kornell, B

    A. Kornell, B. Lindenhovius, and M. Mislove,Quantum CPOs, Proceedings 17th International Confer- ence on Quantum Physics and Logic (2021)

    work page 2021

  30. [30]

    ,A category of quantum posets, Indagationes Mathematicae33(2022), 1137–1171. 77

    work page 2022

  31. [31]

    ,Categories of quantum cpos, preprint (2024)

    work page 2024

  32. [32]

    6, 1137–1171

    ,A category of quantum posets, Indagationes Mathematicae33(2022), no. 6, 1137–1171

    work page 2022

  33. [33]

    Kuperberg and N

    G. Kuperberg and N. Weaver,A Von Neumann Algebra Approach to Quantum Metrics: Quantum Relations, Memoirs of the American Mathematical Society, American Mathematical Society, 2012

    work page 2012

  34. [34]

    A. Kurz, A. Moshier, and A. Jung,Stone duality for relations, Samson abramsky on logic and structure in computer science and beyond, 2023, pp. 159–215

    work page 2023

  35. [35]

    James Laird,Weighted models for higher-order computation, Information and Computation275(2020), 104645

    work page 2020

  36. [36]

    Mac Lane and I

    S. Mac Lane and I. Moerdijk,Sheaves in geometry and logic: A first introduction to topos theory, Universitext, Springer New York, New York, NY, 1992

    work page 1992

  37. [37]

    Murphy,C*-Algebras and Operator Theory, Academic Press, 1990

    G.J. Murphy,C*-Algebras and Operator Theory, Academic Press, 1990

    work page 1990

  38. [38]

    Pedersen,Analysis Now, Springer, 1989

    G.K. Pedersen,Analysis Now, Springer, 1989

    work page 1989

  39. [39]

    Stubbe,Categorical structures enriched in a quantaloid: categories, distributors and functors, Theory and Applications of Categories14(2005), no

    I. Stubbe,Categorical structures enriched in a quantaloid: categories, distributors and functors, Theory and Applications of Categories14(2005), no. 1, 1–45

    work page 2005

  40. [40]

    Takesaki,Theory of operator algebra i, Springer, 2000

    M. Takesaki,Theory of operator algebra i, Springer, 2000

    work page 2000

  41. [41]

    Treves,Topological vector spaces, distributions and kernels, Dover Publications, 2006

    work page 2006

  42. [42]

    Weaver,Quantum relations, Mem

    N. Weaver,Quantum relations, Mem. Amer. Math. Soc.215(2012)

    work page 2012

  43. [43]

    N. E. Wegge-Olsen,K-theory and C*-algebras, Oxford University Press, 1993

    work page 1993

  44. [44]

    Weidmann,Linear Operators in Hilbert Spaces, Springer-Verlag New York Inc., 1980

    J. Weidmann,Linear Operators in Hilbert Spaces, Springer-Verlag New York Inc., 1980. 78

    work page 1980