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arxiv: 2401.17494 · v3 · submitted 2024-01-30 · 🧮 math.CT

Premonoidal and Kleisli double categories

Bojana Femi\'c This is my paper

Pith reviewed 2026-05-24 04:09 UTC · model grok-4.3

classification 🧮 math.CT
keywords premonoidal double categoriesKleisli double categoriesdouble monadsvertical strengthshorizontal strengthspure centerpseudodouble quasi-functorfunny product
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The pith

A premonoidal double category admits a monoidal structure precisely when it is purely central and its binoidal structure comes from a pseudodouble quasi-functor.

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

The paper defines premonoidal double categories as a double-categorical version of premonoidal bicategories and equips them with a closed funny monoidal structure via a funny product and a funny multicategory. It proves that a premonoidal double category is purely central if and only if its binoidal structure is given by a pseudodouble quasi-functor if and only if it admits a monoidal structure. For such categories the monoidal structure extends to the pure center, and the paper also discusses one-sided and general centers. Using companion-lifting properties of vertical structures into horizontal counterparts, the paper shows that vertical strengths on vertical double monads induce horizontal strengths on horizontal double monads, that horizontal strengths correspond to extensions of the canonical action, and that bistrong vertical double monads make their Kleisli double categories premonoidal.

Core claim

A premonoidal double category Dd is purely central if and only if its binoidal structure is given by a pseudodouble quasi-functor if and only if it admits a monoidal structure. For such Dd the monoidal structure extends to the pure center. Vertical strengths on vertical double monads induce horizontal strengths, which correspond one-to-one with extensions of the canonical action of the double category on itself. For a bistrong vertical double monad the corresponding Kleisli double category is premonoidal.

What carries the argument

The equivalence among purely central premonoidal double categories, binoidal structures given by pseudodouble quasi-functors, and monoidal structures, together with companion-lifting properties that relate vertical and horizontal structures.

Load-bearing premise

The companion-lifting properties of vertical structures into their horizontal counterparts hold for the double category under consideration.

What would settle it

A concrete premonoidal double category that is purely central but does not admit a monoidal structure, or whose binoidal structure is not given by a pseudodouble quasi-functor.

read the original abstract

We give a double categorical version of the recently introduced notion of premonoidal bicategories. We introduce a funny product and a funny type of multicategory on double categories granting them a closed funny monoidal structure. We investigate relations between various funny type of structures and premonoidal double categories. We prove that a premonoidal double category $\Dd$ is purely central if and only if its binoidal structure is given by a pseudodouble quasi-functor (a multimap for a Gray type of multicategory) if and only if it admits a monoidal structure. For such $\Dd$ we introduce pure center and show that the monoidal structure on $\Dd$ extends to it. We also discuss one-sided and general centers. Exploiting the companion-lifting properties of vertical structures in a double category into their horizontal counterparts, we prove a series of further results simplifying proofs for the corresponding bicategorical findings. We introduce vertical strengths on vertical double monads and horizontal strengths on horizontal double monads and prove that the former induce the latter. We show that vertical strengths induce actions of the induced horizontally monoidal double category on the corresponding Kleisli double category of the induced horizontal double monad. We prove that there is a 1-1 correspondence between horizontal strengths and extensions of the canonical action of the double category on itself. Finally, we show that for a bistrong vertical double monad the corresponding Kleisli double category is premonoidal.

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 introduces premonoidal double categories as a double-categorical analogue of premonoidal bicategories, along with a 'funny product' and associated multicategory structures on double categories. It proves that a premonoidal double category D is purely central if and only if its binoidal structure arises from a pseudodouble quasi-functor if and only if D admits a monoidal structure, with the monoidal structure extending to the pure center. Further results include vertical strengths on vertical double monads inducing horizontal strengths, actions on Kleisli double categories, a bijection between horizontal strengths and extensions of the canonical action, and that bistrong vertical double monads yield premonoidal Kleisli double categories. The proofs exploit companion-lifting of vertical structures to horizontal counterparts to simplify bicategorical arguments.

Significance. If the central equivalences and inductions hold, the work supplies a coherent double-categorical framework for premonoidal structures and monad strengths, with the companion-lifting technique offering a systematic way to relate vertical and horizontal data. The explicit 1-1 correspondence between strengths and action extensions, together with the Kleisli premonoidality result, are concrete contributions that could support further development in double-category monad theory.

major comments (2)
  1. [Section on purely central premonoidal double categories and the main equivalence theorem (around the statement of the 'D] The central equivalence chain (purely central ⇔ binoidal via pseudodouble quasi-functor ⇔ admits monoidal structure) and the extension to the pure center rest on companion-lifting of vertical structures into horizontal ones. The manuscript invokes this property to simplify bicategorical proofs and relate vertical/horizontal strengths, but does not explicitly verify that the newly defined premonoidal axioms (and the funny product) guarantee the existence or preservation of the required companions for a general premonoidal double category that is not already monoidal. This verification is load-bearing for the iff statements.
  2. [Sections on vertical/horizontal strengths and Kleisli double categories] The induction that vertical strengths on vertical double monads induce horizontal strengths (and the subsequent action on the Kleisli double category) likewise relies on the same companion-lifting step. No separate check is supplied that the premonoidal structure on the Kleisli object preserves the companions needed for the induction.
minor comments (2)
  1. [Introduction and definitions] Notation for the funny product and the pseudodouble quasi-functor could be introduced with a short comparison table to the corresponding bicategorical notions to aid readability.
  2. [Section on horizontal strengths] The abstract claims a '1-1 correspondence' between horizontal strengths and extensions of the canonical action; the precise statement of this bijection (including what data is fixed) should be restated in the body with a reference to the relevant proposition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for highlighting the reliance on companion-lifting in the premonoidal setting. The comments are well-taken; we will revise the manuscript to supply the missing explicit verifications. Point-by-point responses follow.

read point-by-point responses

  1. Referee: [Section on purely central premonoidal double categories and the main equivalence theorem (around the statement of the 'D] The central equivalence chain (purely central ⇔ binoidal via pseudodouble quasi-functor ⇔ admits monoidal structure) and the extension to the pure center rest on companion-lifting of vertical structures into horizontal ones. The manuscript invokes this property to simplify bicategorical proofs and relate vertical/horizontal strengths, but does not explicitly verify that the newly defined premonoidal axioms (and the funny product) guarantee the existence or preservation of the required companions for a general premonoidal double category that is not already monoidal. This verification is load-bearing for the iff statements.

    Authors: We agree that the manuscript does not contain an explicit check that the premonoidal axioms and funny product preserve companions. In the revision we will add a dedicated lemma (placed immediately before the equivalence theorem) proving that every vertical morphism in a premonoidal double category admits a companion and that the binoidal functors, associators and funny product preserve these companions, thereby justifying the lifting to horizontal structures and supporting the full equivalence chain. revision: yes

  2. Referee: [Sections on vertical/horizontal strengths and Kleisli double categories] The induction that vertical strengths on vertical double monads induce horizontal strengths (and the subsequent action on the Kleisli double category) likewise relies on the same companion-lifting step. No separate check is supplied that the premonoidal structure on the Kleisli object preserves the companions needed for the induction.

    Authors: We accept that a separate verification for the Kleisli construction is required. The revised manuscript will include a short proposition showing that the premonoidal structure induced on the Kleisli double category inherits companion-lifting from the base double category (via the standard Kleisli construction for double categories), which legitimises the induction of horizontal strengths and the induced action. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claims rest on new definitions and standard double-category axioms

full rationale

The paper defines premonoidal double categories, funny products, pseudodouble quasi-functors, pure centers, and related structures from scratch, then proves the central equivalence (purely central ⇔ binoidal via pseudodouble quasi-functor ⇔ admits monoidal structure) and the extension to the pure center by direct use of companion-lifting, which is stated as an existing property of double categories rather than derived from the new premonoidal axioms. No equations reduce a 'prediction' to a fitted parameter by construction, no self-citation chains are load-bearing, and no ansatz is smuggled via prior work. The derivation is self-contained against the introduced definitions and standard axioms; the companion-lifting step is invoked as an external tool without evidence of circular dependence on the paper's own structures.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 3 invented entities

The work rests on the standard axioms of categories, double categories, and bicategories together with newly coined definitions; no numerical parameters are fitted and no new physical or computational entities are postulated.

axioms (1)
  • standard math Standard axioms of (double) categories and bicategories as developed in prior literature
    All new structures are defined by extending these background axioms.
invented entities (3)
  • funny product no independent evidence
    purpose: Equips double categories with a closed funny monoidal structure
    Newly defined operation introduced to obtain the monoidal structure.
  • premonoidal double category no independent evidence
    purpose: Double-categorical analogue of premonoidal bicategories
    Central new object whose properties are investigated.
  • pure center no independent evidence
    purpose: Extension of the monoidal structure for purely central premonoidal double categories
    Newly introduced center construction.

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Works this paper leans on

34 extracted references · 34 canonical work pages · 1 internal anchor

  1. [1]

    Bourke, N

    J. Bourke, N. Gurski, The Gray T ensor Product Via Factorisation, Appl. Categ. Struc- tutes 25 /4 (2017), 603–624. 2, 14

    work page 2017

  2. [2]

    B ¨ ohm,The Gray Monoidal Product of Double Categories , Appl

    G. B ¨ ohm,The Gray Monoidal Product of Double Categories , Appl. Categ. Structures 28 (2020), 477–515. https://doi.org/10.1007/s10485-019-09587-5

    work page doi:10.1007/s10485-019-09587-5 2020

  3. [3]

    Brown, C

    R. Brown, C. B. Spencer, Double groupoids and crossed modules , Cahiers Topologie G´ eom. Diff´ erentielle,17/4 (1976), 343–362

    work page 1976

  4. [4]

    Cruttwell, M

    G. Cruttwell, M. Shulman, A unified framework for generalized multicategories, Theory Appl. Categ. 24 (2010), 580–655

    work page 2010

  5. [5]

    P .F. Faul, G. Manuell, J. Siqueira, 2-Dimensional bifunctor theorems and distributive laws, Theory Appl. Categ. 37 (2021), 1149–1175

    work page 2021

  6. [6]

    Femi´ c,Bifunctor Theorem and strictification tensor product for do uble categories with lax double functors, Theory Appl

    B. Femi´ c,Bifunctor Theorem and strictification tensor product for do uble categories with lax double functors, Theory Appl. Categ. 39 (2023), 824–873

    work page 2023

  7. [7]

    Femi´ c, Enrichment and internalization in tricategories, the case of tensor categories and alternative notion to intercategories , Filomat 38/8, 2601–2660

    B. Femi´ c, Enrichment and internalization in tricategories, the case of tensor categories and alternative notion to intercategories , Filomat 38/8, 2601–2660

    work page

  8. [8]

    Femi´ c, Gray (skew) multicategories for double and Gray-categorie s, arxiv .org/abs/2408.00561

    B. Femi´ c, Gray (skew) multicategories for double and Gray-categorie s, arxiv .org/abs/2408.00561

    work page arXiv

  9. [9]

    Femi´ c, S

    B. Femi´ c, S. Halbig Categorical centers and Y etter–Drinfel‘d-modules as 2-ca tegorical (bi)lax structures, arXiv:2306.05337

    work page arXiv

  10. [10]

    Gambino, R

    N. Gambino, R. Garner, C. Vasilakopoulou, Monoidal Kleisli Bicategories and the Arithmetic Product of Coloured Symmetric Sequences , Doc. Math. 29/3 (2024), 627– 702

    work page 2024

  11. [11]

    Gordon, A

    R. Gordon, A. J. Power, R. Street, Coherence for tricategories , Memoirs of the Amer. Math. Soc. 117/558 (1995), 19, 28

    work page 1995

  12. [12]

    Grandis, Higher Dimensional Categories: From Double to Multiple Cat egories, World Scientific (2019)

    M. Grandis, Higher Dimensional Categories: From Double to Multiple Cat egories, World Scientific (2019). 133

    work page 2019

  13. [13]

    Grandis, R

    M. Grandis, R. Par´ e, Limits in double categories , Cahiers Topol. G´ eom. Diff. Cat´ eg. 40 (1999), 162–220

    work page 1999

  14. [14]

    Grandis, R

    M. Grandis, R. Par´ e,Adjoint for double categories, Cahiers de Topologie et G´ eom´ etrie Diff´ erentielle Cat´ egoriques45/3 (2004), 193–240

    work page 2004

  15. [15]

    J. W. Gray , Formal category theory: adjointness for 2-categories , Lecture Notes in Mathematics 391, Springer-V erlag, Berlin-New York (1974) 1, 19, 27

    work page 1974

  16. [16]

    Hermida, Representable multicategories, Adv

    C. Hermida, Representable multicategories, Adv . Math.151/2 (2000), 164–225

    work page 2000

  17. [17]

    Marmolejo, Doctrines whose structure forms a fully faithful adjoint st ring, Theory and Applications of Categories 3/2 (1997), 23–24

    F. Marmolejo, Doctrines whose structure forms a fully faithful adjoint st ring, Theory and Applications of Categories 3/2 (1997), 23–24

    work page 1997

  18. [18]

    McDermott, T

    D. McDermott, T. Uustalu, What makes a strong monad? , Electronic Proceedings in Theoretical Computer Science 360 (2022), 113–133

    work page 2022

  19. [19]

    E. Meir, M. Szymik, Drinfeld centers for bicategories , Doc. Math. 20 (2015), 707–735

    work page 2015

  20. [20]

    Møgelberg, S

    R. Møgelberg, S. Staton, Linear usage of state, Logical Methods in Computer Science 10/1 (2014)

    work page 2014

  21. [21]

    Moggi, Computational lambda-calculus and monads , In Proceedings, Fourth An- nual Symposium on Logic in Computer Science

    E. Moggi, Computational lambda-calculus and monads , In Proceedings, Fourth An- nual Symposium on Logic in Computer Science. IEEE Comput. So c. Press (1989)

    work page 1989

  22. [22]

    Moggi, Notions of computation and monads , Information and Computation 93/1 (1991), 55–92

    E. Moggi, Notions of computation and monads , Information and Computation 93/1 (1991), 55–92

    work page 1991

  23. [23]

    Paquet, P

    H. Paquet, P . Saville, Effectful semantics in 2-dimensional categories: pre- monoidal and Freyd bicategories , Electronic Proceedings in Theoretical Com- puter Science 397/3 (2023), 190–209 (Proceedings of the Sixth International Conference on Applied Category Theory 2023), Open Publishi ng Association, https://doi.org/10.4204/EPTCS.397.12

    work page doi:10.4204/eptcs.397.12 2023

  24. [24]

    A Nominal Approach to Probabilistic Separation Logic,

    H. Paquet, P . Saville, Effectful semantics in 2-dimensional categories: strong, com - mutative, and concurrent pseudomonads , LICS ’24: Proceedings of the 39th An- nual ACM /IEEE Symposium on Logic in Computer Science Article 61, 1 – 15, https://doi.org/10.1145/3661814.3662130

    work page doi:10.1145/3661814.3662130

  25. [25]

    Paquet, P

    H. Paquet, P . Saville, Strong pseudomonads and premonoidal bicategories , arXiv:2304.11014 (choose v1)

    work page arXiv

  26. [26]

    Power, Premonoidal categories as categories with algebraic struc ture, Theoretical Computer Science 278/(1-2) (2002), 303–321

    J. Power, Premonoidal categories as categories with algebraic struc ture, Theoretical Computer Science 278/(1-2) (2002), 303–321

    work page 2002

  27. [27]

    Power, E

    J. Power, E. Robinson, Premonoidal categories and notions of computation, Math. Struc- tures Comput. Sci. 7/5 (1997), 453–468

    work page 1997

  28. [28]

    Power, H

    J. Power, H. Thielecke, Closed Freyd- and κ-categories, Automata, Languages and Programming, Springer Berlin Heidelberg (1999), 625–634

    work page 1999

  29. [29]

    private communication with Philip Saville. 134

    work page

  30. [30]

    Constructing symmetric monoidal bicategories

    M. Shulman, Constructing symmetric monoidal bicategories , arXiv: 1004.0993

    work page internal anchor Pith review Pith/arXiv arXiv

  31. [31]

    Shulman Framed bicategories and monoidal fibrations , Theory Appl

    M. Shulman Framed bicategories and monoidal fibrations , Theory Appl. Categ. 20/18 (2008), 650–738

    work page 2008

  32. [32]

    Staton, Commutative semantics for probabilistic programming , In Programming Languages and Systems, Springer Berlin Heidelberg (2017), 855–879

    S. Staton, Commutative semantics for probabilistic programming , In Programming Languages and Systems, Springer Berlin Heidelberg (2017), 855–879

    work page 2017

  33. [33]

    Thielecke, Continuation semantics and self-adjointness , Electronic Notes in Theo- retical Computer Science 6 (1997), 348–364

    H. Thielecke, Continuation semantics and self-adjointness , Electronic Notes in Theo- retical Computer Science 6 (1997), 348–364

    work page 1997

  34. [34]

    Uustalu, V

    T. Uustalu, V . V ene, Comonadic Notions of Computation , Electronic Notes in Theo- retical Computer Science 203 (2008), 263–284. 135

    work page 2008