Bicategories of algebras for relative pseudomonads

Andrew Slattery; Nathanael Arkor; Philip Saville

REVIEW 2 major objections 4 minor 3 cited by

Bicategories of algebras for relative pseudomonads

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read For every relative pseudomonad, its pseudoalgebras form a terminal resolution, and the Kleisli bicategory embeds into them.

desk verdict Important and mostly sound; the cocompleteness characterization rests on a delicate coherence chase that needs careful checking. read the letter →

arxiv 2501.12510 v1 pith:75HUSOKN submitted 2025-01-21 math.CT

Nathanael Arkor , Philip Saville , Andrew Slattery This is my paper · ORCID

classification math.CT MSC 18C1518C2018D6018M5018N1018N1518N20

keywords relativepseudomonadpseudoalgebraKleislibicategorypresheafconstructioncocompletecategorydistributorcoherencetheoremlax-idempotent

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 builds a two-dimensional algebra theory for relative pseudomonads, which are monad-like structures that need not be endofunctors, the motivating example being the presheaf construction on small categories. It defines pseudoalgebras for such a T and constructs the free–forgetful relative pseudoadjunction from the bicategory PsAlg(T) of T-pseudoalgebras, proving it is 2-terminal among all resolutions of T: every relative pseudoadjunction inducing exactly T factors through it, with the mediating pseudofunctor unique up to isomorphism. Consequently, the Kleisli bicategory embeds fully faithfully into PsAlg(T) as the free pseudoalgebras, and when the codomain is a 2-category this yields a coherence theorem: Kl(T) is biequivalent to the full sub-2-category of free pseudoalgebras. For the presheaf relative pseudomonad, the paper proves that pseudoalgebras are exactly locally small categories with a choice of small colimits, so the bicategory of distributors is biequivalent to the full sub-2-category of presheaf categories inside the 2-category of cocomplete categories. Along the way it extends doctrinal adjunction, transport of structure, and lax-idempotence to the relative setting.

What carries the argument

The central object is a T-pseudoalgebra for a J-relative pseudomonad T: an object A of the codomain together with an extension operator (−)^a : E[JX,A] → E[TX,A] and invertible 2-cells â and ã satisfying associativity and unit laws (Definition 3.1). The load-bearing mechanism is the resolution 2-category Res(T), whose objects are relative pseudoadjunctions inducing exactly T, together with the two universal constructions: the Kleisli bicategory (bi-initial, Theorem 6.3) and the pseudoalgebra bicategory (2-terminal, Theorem 6.10). The comparison pseudofunctor IT : Kl(T) → PsAlg(T) of Corollary 6.11 is what converts the coherence statement: it is fully faithful and, when E is a 2-category, exhibits Kl(T) as biequivalent to the full sub-2-category of free pseudoalgebras (Corollary 6.15). For the presheaf application, the pivotal identity is that a functor D → A with small domain D has a colimit exactly when it extends along the free cocone inclusion D ↪ D^⊤, and Lemma 7.8 uses the pseudoalgebra structure to construct that extension, thereby building colimits in any pseudoalgebra.

What would settle it

Attempt to construct a P-pseudoalgebra structure on a locally small category that lacks some small colimit; the pseudoalgebra axioms require a left extension of every functor from a small category, and if such an extension can be defined despite the missing colimit, Theorem 7.14 is false. Conversely, the theorem's proof constructs the missing colimit from the extension operator, so checking this single example settles the matter.

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Extended reading notes

Core claim

Theorem 6.10 states that for any relative pseudomonad T, the free–forgetful relative pseudoadjunction from the bicategory PsAlg(T) of T-pseudoalgebras and pseudomorphisms is 2-terminal in the 2-category Res(T) of resolutions of T: every relative pseudoadjunction inducing exactly T has a unique morphism to the pseudoalgebra resolution, and the only 2-cell from that morphism to itself is the identity. Consequently (Corollary 6.11) there is a unique fully faithful pseudofunctor Kl(T) → PsAlg(T) whose image is the free pseudoalgebras, and when the codomain is a 2-category this exhibits Kl(T) as biequivalent to the full sub-2-category of free pseudoalgebras (Corollary 6.15). In the specific case of the presheaf relative pseudomonad P, the paper proves that every P-pseudoalgebra is precisely a locally small category equipped with chosen small colimits (Theorem 7.14), so the bicategory Dist of distributors is biequivalent to the full sub-2-category of the 2-category COC spanned by presheaf categories. The proof of Theorem 7.14 constructs colimits inside any pseudoalgebra using the extension operator and the one-point-cocone category D^⊤, and it relies on Lemma 7.8, which assumes D^⊤ lies in the domain and that the unit preserves terminal objects; as Remark 7.20 notes, the technique is not known to extend to enriched categories.

Load-bearing premise

The identification of free-cocompletion pseudoalgebras with cocomplete categories (Theorem 7.14) rests on the assumption that every relevant diagram shape D and its one-point-cone extension D^⊤ belong to the domain of the relative pseudomonad, and that the unit preserves terminal objects; this holds for ordinary categories but the authors state they do not know how to extend it to enriched categories.

Editorial extensions

If this is right

For every relative pseudomonad T, the bicategory PsAlg(T) of T-pseudoalgebras is the terminal resolution of T: every relative pseudoadjunction inducing exactly T factors through it, and the mediating pseudofunctor is unique up to unique isomorphism (Theorem 6.10).
The Kleisli bicategory Kl(T) embeds fully faithfully into PsAlg(T) as the free pseudoalgebras (Corollary 6.11); when the codomain is a 2-category, Kl(T) is biequivalent to the full sub-2-category of free pseudoalgebras (Corollary 6.15).
For the presheaf relative pseudomonad P, the pseudoalgebras are precisely locally small categories equipped with chosen small colimits, so the bicategory Dist of distributors is biequivalent to the full sub-2-category of presheaf categories in the 2-category of cocomplete categories (Theorem 7.14 and Corollary 6.15).
For any class Φ of small categories closed under adjoining a terminal object, the free Φ-cocompletion relative pseudomonad has exactly the Φ-cocomplete categories as pseudoalgebras, and every pseudoalgebra is lax-idempotent (Theorem 7.14, Corollary 7.13).
Monads relative to the free Φ-cocompletion of a small category A are equivalent to Φ-cocontinuous monads on Φ(A), and the equivalence commutes with the taking of algebras (Theorem 8.3).

Reading between the lines

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

Beyond the paper's claims: the 2-terminality theorem gives a uniform strictification recipe, so any Kleisli bicategory for a relative pseudomonad valued in a 2-category is automatically biequivalent to the concrete 2-category of free algebras; other size-sensitive constructions (free coproducts, Ind-completions, sifted colimits) should admit the same coherence theorem without case-by-case work.
The limitation flagged in Remark 7.20 suggests an enriched analogue will need a different technique than the D^⊤-cocone construction; if developed, it would likely give enriched coherence theorems for enriched distributors and cocomplete enriched categories.
One testable extension: the stronger notion of lax-idempotent relative pseudomonad (all pseudoalgebras lax-idempotent) may be the right one for future work; if it implies every colax morphism between pseudoalgebras is pseudo, it would fill the gap noted in Remark 5.24.
The correspondence of Theorem 8.3 between relative monads and cocontinuous monads is likely an instance of a more general relative monadicity that could be checked by verifying the triangle comparison for Ind-completions and sifted colimits directly.

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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

Bottom line: this is a serious and significant paper. It supplies the missing theory of pseudoalgebras for relative pseudomonads, and the central universal properties—bi-initiality of the Kleisli resolution and 2-terminality of the pseudoalgebra resolution—are new, well-motivated, and, as far as I can check, correctly proven. Theorem 6.10 is particularly solid: the proof uses local faithfulness of the forgetful pseudofunctor plus the canonical pseudoalgebra structure on right pseudoadjoints, and it does not borrow from the later cocompleteness material. Corollary 6.11, the fully faithful embedding of the Kleisli bicategory into PsAlg(T), is a nice coherence result that stands alone.

The paper gives credit where it is due: the comparison with Marmolejo–Wood no-iteration pseudoalgebras (Section 3.2) is careful, and the authors are explicit about what is not known (Remark 5.24, Remark 7.20). The citation pattern is normal for the subject; the heavy reliance on FGHW18 is justified because that paper is the source of the relative pseudomonad formalism.

The soft spot is exactly where the stress-test places it: Lemma 7.8. The uniqueness part of the colimit construction evaluates the large diagrams (31) and (32) at the terminal object of T(D), and concludes that a composite morphism is the identity from the universal property of that terminal object. The chase is intricate and mixes the pseudoalgebra coherence cells with pseudonaturality of the extension operator; the text does not fully spell out how the pseudonaturality 2-cells compose along the evaluation. I did not find a definite error, but I also could not verify every step from the text alone. Consequently, Theorem 7.14—the characterization of pseudoalgebras for free cocompletions as the Φ-cocomplete categories—is the least secure pillar. If Lemma 7.8 has a hidden coherence bug, the distributors–presheaf biequivalence would lose its proof, though the general terminality theorem and the Kleisli embedding would survive.

The Section 8 application (relative monads as cocontinuous monads) is a nice payoff but inherits the dependence on Theorem 7.14. For a reader, the value is mainly in Sections 3–6, which lay out a solid framework; Section 7 should be read with extra care.

This paper deserves a serious referee. The right referee will spend most of their time on Lemma 7.8, ideally redoing the uniqueness chase or at least testing it on the free cocompletion example. I would engage with it and recommend the same.

Referee Report

2 major / 4 minor

Summary. The paper develops a theory of pseudoalgebras for relative pseudomonads. It defines bicategories of pseudoalgebras with strict, pseudo, lax, and colax morphisms; constructs a free--forgetful relative pseudoadjunction; proves doctrinal adjunction and transport of structure; extends lax-idempotence to the relative setting; and establishes universal properties for the Kleisli and pseudoalgebra resolutions of a relative pseudomonad, with the pseudoalgebra resolution shown to be 2-terminal. The main applications are a characterisation of pseudoalgebras for free cocompletion relative pseudomonads as categories with the appropriate class of colimits (Theorem 7.14), yielding a proof that the bicategory of distributors is biequivalent to the 2-category of presheaf categories, and a correspondence between cocompletion-relative monads and cocontinuous monads on free cocompletions (Theorem 8.3). The paper is carefully written, with extensive diagrammatic proofs and explicit acknowledgement of its own open points.

Significance. If the central results hold, the paper fills a genuine gap in two-dimensional monad theory: relative pseudomonads had no systematic algebra theory, and the paper supplies one, including the expected universal properties and the first rigorous proof, to the authors' knowledge, of the folklore biequivalence between distributors and presheaf categories. The treatment of lax-idempotence is notably more subtle than in the non-relative case, and the authors are honest about the limits of their techniques, for example in Remark 5.24 and Remark 7.20. The paper does not rely on machine-checked proofs, but its diagrammatic arguments are detailed and the main structure is coherent. The principal reservation is that the characterisation in Theorem 7.14 depends on Lemma 7.8, a long coherence argument whose key uniqueness step is not fully supported as written.

major comments (2)

[§7.1, Lemma 7.8 (proof, diagrams (31)–(32))] The uniqueness argument in Lemma 7.8 is load-bearing for Theorem 7.14, and as written it is not fully supported. The proof evaluates diagram (31) at the terminal object t of T(D) and then pastes on diagram (32); however, in (32) the left vertical arrow is evaluated at i_{D^⊤}(⊤), an object of T(D^⊤), while t is an object of T(D). It is not immediately evident that these objects can be identified, and the text gives no explicit justification for doing so. If they are not identified, the clockwise composite whose equality with (˜a^{−1}_f)^a(t) is claimed is not even well-formed. Please either repair this step or expand the diagram chase so that the identification of the two objects and the composition of the pseudonaturality 2-cells along the evaluation at t is spelled out. Because Lemma 7.8 is the only route by which the paper obtains colimits in arbitrary pseudoalgebras in Theorem 7.14, this is not a merely stylistic issue.
[§7.2, Theorem 7.14 (dependency chain)] The proof of Theorem 7.14 chains Lemma 7.8 through Proposition 7.12 and Corollary 7.13 into Corollary 5.8 to prove that the comparison pseudofunctor is locally an equivalence. This is structurally sound, but it means that local full faithfulness of Φ-COC → PsAlg(Φ) is obtained only after Lemma 7.8 has been fixed. Please make this dependency explicit in the proof, and state explicitly that Corollary 7.13 uses Proposition 7.12, and hence Lemma 7.8, at the point where it asserts that fa is the left extension of f. A reader who is not prepared to accept Lemma 7.8 as a black box currently has no way to isolate the rest of the argument.

minor comments (4)

[§7.1, proof of Lemma 7.8] The notation J(D^⊤) = D^⊤ = (JD)^⊤ is used without comment. Since J is injective on objects, the identification is harmless, but it should be stated explicitly before the diagram chase begins.
[§7.1, proof of Lemma 7.8] The labels 'A unit.' and 'A unit.′' appear in the diagrams without definitions. Please give the precise displayed axiom being invoked, for instance part (6) of Definition 3.1 or Lemma 3.15, so that the reader can check the two labels separately.
[§7.2, Theorem 7.14 and Remark 7.20] Since Theorem 7.14 is the flagship application and Remark 7.20 explicitly limits the technique to ordinary categories, the statement of Theorem 7.14 should carry a cross-reference to Remark 7.20 and should state clearly that the enriched case is outside the scope of the proof.
[§6.1, Definition 6.1 and Remark 6.8] The definition of Res(T) uses strict equalities MLX = L′X and R = R′M. This is internally consistent, but the words 'bi-initial' and '2-terminal' may suggest a weaker bicategorical universal property. Remark 6.8 is helpful; consider adding a caution immediately after Definition 6.1 that the universal properties are relative to this strict class of resolution morphisms.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorems are derived from explicit constructions and internal proofs, with self-citations confined to peripheral or background support.

full rationale

The paper's central derivation chain is self-contained rather than circular. Theorem 6.10 proves 2-terminality of the pseudoalgebra resolution by explicitly constructing canonical pseudoalgebra structure on the right pseudoadjoint of any resolution (Lemma 6.9) and using local faithfulness of the forgetful pseudofunctor; the conclusion is not assumed in the construction. The Kleisli embedding (Corollary 6.11) is then a formal consequence of the universal properties, not a restatement of an input. The main application, Theorem 7.14, is a genuine characterization: Proposition 7.12 constructs colimits in arbitrary pseudoalgebras using the internally proved Lemma 7.8, and the converse direction is obtained from the comparison pseudofunctor supplied by Theorem 6.10 applied to the free cocompletion resolution, rather than from an assumption that cocomplete categories are pseudoalgebras. No fitted parameter is renamed as a prediction, and no definition is formulated in terms of the result it is used to prove. The cited prior work [FGHW18] provides background and foundational definitions but is not authored by the present authors; the authors' own citations, such as [AM24] and [AM25], support peripheral applications and are published independent results rather than load-bearing self-justification. Remark 7.20 honestly records a limitation of the technique in the enriched setting; this is an internal scope or correctness concern, not evidence of circularity. The intricate uniqueness argument inside Lemma 7.8 is a possible correctness risk, but it is an internal proof step, not a reduction of the theorem to its own assumptions. I therefore find no significant circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central results do not introduce numeric parameters. They rely on standard categorical infrastructure (coherence, Kan extensions, Yoneda) and on the assumed data of a relative pseudomonad. No new postulates with independent empirical handles are introduced.

assumptions (7)

standard math Bicategorical coherence theorem: pasting diagrams can be elided and diagrams commuting by coherence need not be labelled.
Invoked throughout Section 2.1 and proof of Theorem 3.9 to validate unlabelled diagrams.
standard math Yoneda lemma and density of the Yoneda embedding for free cocompletions.
Used in Lemma 7.9 and Theorem 7.14 to identify terminal objects with colimits of dense functors.
standard math Existence of left Kan extensions in CAT along Yoneda embeddings.
Used in Example 2.3 and Section 7 to define extension operators for presheaf pseudomonads.
domain assumption Set-theoretic size distinction between small and locally small categories, with a chosen universe.
The presheaf construction is defined only on small categories; the paper relies on the inclusion Cat into CAT and the existence of presheaf categories as locally small categories (Section 2.3, Example 2.8).
domain assumption J : A -> E is a pseudofunctor between bicategories and T is a J-relative pseudomonad.
All definitions and theorems are parameterized by such J and T; results hold for this class of inputs.
domain assumption For Theorem 7.14, Phi and Phi' are classes of small categories with D in Phi implying D^top in Phi', and J is the inclusion of Phi' into CAT.
Lemma 7.8 and Proposition 7.12 require D^top in the domain and that the unit of the free cocompletion preserves the terminal object.
domain assumption For Lemma 7.8, A is a full sub-2-category of CAT, T(D) has a terminal object, and i_{D^top} preserves the terminal object.
These are the explicit hypotheses of the technical colimit-construction lemma.

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Pith. "Pith review of Bicategories of algebras for relative pseudomonads." pith.science (2026). https://pith.science/paper/75HUSOKN

@misc{pith2026250112510,
  author       = {Pith},
  title        = {Pith review of: Bicategories of algebras for relative pseudomonads},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/75HUSOKN}},
  note         = {Machine review of arXiv:2501.12510}
}

abstract

We introduce pseudoalgebras for relative pseudomonads and develop their theory. For each relative pseudomonad $T$, we construct a free--forgetful relative pseudoadjunction that exhibits the bicategory of $T$-pseudoalgebras as terminal among resolutions of $T$. The Kleisli bicategory for $T$ thus embeds into the bicategory of pseudoalgebras as the sub-bicategory of free pseudoalgebras. We consequently obtain a coherence theorem that implies, for instance, that the bicategory of distributors is biequivalent to the 2-category of presheaf categories. In doing so, we extend several aspects of the theory of pseudomonads to relative pseudomonads, including doctrinal adjunction, transport of structure, and lax-idempotence. As an application of our general theory, we prove that, for each class of colimits $\Phi$, there is a correspondence between monads relative to free $\Phi$-cocompletions, and $\Phi$-cocontinuous monads on free $\Phi$-cocompletions.

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Forward citations

Cited by 3 Pith papers

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A monadic DSL yields correct-by-construction, substitution-stable bidirectional elaborators for Martin-Löf type theory that extract algebraically from a presheaf model.
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In virtual equipments with mild assumptions, presheaf constructions provide free cocompletions under classes of weights, extending weighted colimits and enabling constructions for enriched categories in monoidal categ...
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math.CT 2025-06 conditional novelty 7.0 of 10

Left oplax Kan extensions turn relative 2-monads into pseudomonads with the same colax algebras, producing the weak ultracompletion pseudomonad for ultracategories.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages · cited by 3 Pith papers

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

‘Monads need not be endofunctors’

[ACU10] Thorsten Altenkirch, James Chapman and Tarmo Uustalu. ‘Monads need not be endofunctors’. In:International Conference on Foundations of Software Science and Computational Structures. Springer. 2010, pp. 297–311 (cit. on pp. 1, 6, 8). [ACU15] Thorsten Altenkirch, James Chapman and Tarmo Uustalu. ‘Monads need not be endofunctors’. In:Logical Methods ...

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[2] [2]

Cambridge University Press, 1994 (cit. on p. 47). [Bou14] John Bourke. ‘Two-dimensional monadicity’. In:Advances in Mathematics252 (2014), pp. 708–747 (cit. on p. 1). [BPG21] Erwan Beurier, Dominique Pastor and René Guitart. ‘Presentations of clusters and strict free-cocompletions’. In:Theory and Applications of Categories36.17 (2021), pp. 492–513 (cit. o...

work page 2014

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arXiv: 2402.11703 (cit. on p. 3). [MP85] Saunders MacLane and Robert Paré. ‘Coherence for bicategories and indexed categor- ies’. In:Journal of Pure and Applied Algebra37 (1985), pp. 59–80 (cit. on p. 42). [MW12] Francisco Marmolejo and Richard J. Wood. ‘Kan extensions and lax idempotent pseudomonads’. In:Theory and Applications of Categories26.1 (2012), ...

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Cambridge University Press, 1982 (cit. on pp. 3, 42, 48, 49). [KL00] Gregory Maxwell Kelly and Stephen Lack. ‘On the monadicity of categories with chosen colimits’. In:Theory and Applications of Categories7.7 (2000), pp. 148–170 (cit. on p. 44). [KL97] Gregory Max Kelly and Stephen Lack. ‘On property-like structures’. In:Theory and Applications of Categor...

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Cambridge University Press, 2013 (cit. on p. 21). [Hyl14] JME Hyland. ‘Elements of a theory of algebraic theories’. In:Theoretical Computer Science 546 (2014), pp. 132–144 (cit. on p. 2). [IK86] Geun Bin Im and Gregory Max Kelly. ‘A universal property of the convolution monoidal structure’. In:Journal of Pure and Applied Algebra43.1 (1986), pp. 75–88 (cit...

work page 2014

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American Mathematical Society, 1995 (cit. on p. 42). [GU71] Peter Gabriel and Friedrich Ulmer.Lokal präsentierbare kategorien. Springer-Verlag, 1971 (cit. on p. 3). [Gui75] René Guitart. ‘Monades involutives complémentées’. In: Cahiers de topologie et géométrie différentielle16.1 (1975), pp. 17–101 (cit. on p. 2). [Gur13] Nick Gurski.Coherence in three-di...

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American Mathematical Society, 2017 (cit. on pp. 39, 51). [GL12] Richard Garner and Stephen Lack. ‘Lex colimits’. In:Journal of Pure and Applied Algebra 216.6 (2012), pp. 1372–1396 (cit. on p. 3). [GPS95] Robert Gordon, Anthony John Power and Ross Street.Coherence for tricategories. Vol

work page 2012

[8] [2019]

(cit. on pp. 1, 23). [Wal69] Robert Frank Carslaw Walters. ‘An alternative approach to universal algebra’. In: Reports of the Midwest Category Seminar III. Springer. 1969, pp. 64–73 (cit. on p. 8). [Wal70] Robert Frank Carslaw Walters. ‘A categorical approach to universal algebra’. PhD thesis. The Australian National University, 1970 (cit. on p. 8). [Zöb7...

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