Idempotence for relative monads

Andrew Slattery; Nathanael Arkor

REVIEW 2 major objections 3 minor 2 references

Idempotence for relative monads

T0 review · 2 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read For relative monads, idempotence splits into two notions; only the stronger restores the classical theorems.

desk verdict A clean, well-motivated distinction between idempotence and algebraic idempotence for relative monads; the main theorem is solid, but one advertised corollary rests on an unstated external theorem. read the letter →

arxiv 2508.17794 v2 pith:SAAOSVZT submitted 2025-08-25 math.CT

Nathanael Arkor , Andrew Slattery This is my paper · ORCID

classification math.CT MSC 18C1518C20

keywords relativemonadsidempotencealgebraicextensionoperatoradjunctionsreflectivefunctorsfreecocompletionsj-ary

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

This paper isolates a subtlety that appears only for monads relative to a functor, not for ordinary monads on a category. It defines two notions: a j-relative monad is idempotent when its extension operator is invertible, and algebraically idempotent when every T-algebra is idempotent. The first is strictly weaker, as Example 2.9 shows, and the paper's thesis is that the second is the right generalization. The central result, Corollary 3.13, characterizes algebraic idempotence as exactly the condition that the forgetful functor from T-algebras to the ambient category is j-reflective, i.e., fully faithful with a left relative adjoint. For ordinary monads, and for relative monads whose root is a free cocompletion, the two notions coincide, so the subtlety disappears in those settings.

What carries the argument

The load-bearing objects are the two extension operators. A j-relative monad T on a functor j : A→E consists of objects ta∈E, units η_a : ja→ta, and an extension rule f↦f† turning each f:ja→tb into ta→tb, satisfying unit and associativity laws. Its algebras replace the single object by an arbitrary e∈E and the extension rule by f↦f⋊:ta→e. Idempotence asks that the first rule be bijective; algebraic idempotence asks the same for the rules of all algebras. The bridge to category theory is that these bijectivity conditions are equivalent to full faithfulness of forgetful functors: an idempotent relative monad is one whose Kleisli category embeds fully into the image of t, and an algebraically idempotent one is one whose algebra forgetful functor is j-reflective, meaning fully faithful with a left j-relative adjoint. The paper also uses j-ary monads—monads presented by their precomposition with the root—to transfer algebraic idempotence along extensions.

What would settle it

Construct, for a free cocompletion root φ_A : A→Φ(A), a φ_A-relative monad that is idempotent but admits a non-idempotent algebra; such an example would refute Corollary 3.18. More directly, the paper's own Remark 2.10 invites the same construction with a dense root—an explicit dense-root counterexample would show idempotence is strictly weaker even under density.

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Extended reading notes

Core claim

At the paper's center is a distinction that does not exist for ordinary monads. For a j-relative monad T, write †: E(j,t) ⇒ E(t,t) for its extension operator. Idempotence is the statement that † is invertible, equivalently that every morphism ja→tb extends uniquely along ηa. Algebraic idempotence is the stronger statement that every T-algebra (e, ⋊) has invertible extension operator ⋊, equivalently that every object e is η-orthogonal. The paper's main structural result is Corollary 3.13: T is algebraically idempotent exactly when the forgetful functor u_T : Alg(T)→E is j-reflective, which here reduces to full faithfulness since the relative left adjoint exists automatically. This recovers, in the relative setting, the classical characterization of idempotent monads by their algebras, and Corollary 3.18 shows that when the root is a free cocompletion φ_A : A→Φ(A), idempotence and algebraic idempotence coincide. The authors therefore argue that algebraic idempotence is the appropriate notion, with the traditional terminology reserved for it.

Load-bearing premise

The free-cocompletion result, Corollary 3.18, takes as a black box a companion theorem that every φ_A-relative monad extends along the free cocompletion to a φ_A-ary monad; if that extension theorem has hidden hypotheses or fails for some colimit classes, the coincidence of the two notions of idempotence in that setting collapses.

Editorial extensions

If this is right

For ordinary monads, viewed as relative monads on the identity functor, the two notions coincide, so the classical theory of idempotent monads is unchanged.
Algebraic idempotence can be read off from the algebra category: one only needs to check that the forgetful functor Alg(T)→E is fully faithful, without enumerating algebras.
Every idempotent relative monad has its Kleisli category as the initial j-reflective resolution and its idempotent algebras as the terminal one; algebraic idempotence makes the two sides coincide.
For relative monads over free cocompletions—the usual setting for monads presented by operations—the distinction disappears: idempotent implies algebraically idempotent.
In the special case of monads presented in extension form, the characterisations specialise to concrete criteria for idempotence of the extension operator.

Reading between the lines

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

A broader design principle suggests itself: for relative monads, every property defined for the monad should also be defined for its algebras; the paper points to strong relative monads as the next case, where a notion of strong algebra would be needed.
Corollary 3.18 suggests a practical proof strategy: to show a relative monad is algebraically idempotent, extend it to an ordinary monad on the free cocompletion, prove idempotence of that monad, and pull the conclusion back through the j-ary equivalence.
The paper leaves open whether a dense root forces idempotence to imply algebraic idempotence; a positive or negative answer would sharpen the boundary of the phenomenon.
If the pattern extends to relative pseudomonads, as the authors suspect, algebraic lax-idempotence rather than lax-idempotence is likely the correct notion in bicategorical settings such as substitution monoidal structures.

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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

Short version: this is a good paper and worth a serious referee. It introduces a genuinely new distinction—algebraic idempotence vs the existing (non-algebraic) idempotence for relative monads—and proves the characterization that T is algebraically idempotent exactly when Alg(T) -> E is j-reflective (Cor 3.13). The counterexample in Example 2.9, separating the two notions, is simple and convincing. That alone justifies the paper.

What it does well: the proofs are clean, the paper is honest about where the classical equivalences fail, and the Kleisli/reflection results in Section 3 are useful. Proposition 3.12 gives the terminal reflective resolution, and Corollary 3.13 is the payoff. I also like the clear explanation of why algebraic idempotence is the right notion, not just a technical strengthening.

Soft spots, in order of importance.

1. Corollary 3.18 (idempotence = algebraic idempotence for free cocompletion roots) is the advertised special case, and its proof is a chain that imports [ASS25, Theorem 8.3] as a black box. The hypotheses of that theorem aren't stated, and the corollary's conclusion depends on the extended monad being phi_A-ary and exhibiting the left extension (Props 3.16 and 3.17). If the extension theorem has hidden size or pointwise conditions, the equivalence in 3.18 breaks. I don't think this is a fatal flaw—it's normal to cite your own prior work in a companion paper—but the paper should at least state the theorem's hypotheses or flag the dependency. There's also a tension worth a sentence: Remark 2.10 says a dense root might separate the notions, yet a free cocompletion root is dense and 3.18 says they coincide for it. The reader is left to figure out what the expected behaviour is outside the free-cocompletion case.

2. Proposition 3.12's terminality proof is sketched ('goes through in essentially the same way'). Minor—the reader can fill it in, but it should be a real argument.

3. The Setting paragraph asserts the enriched and virtual-equipment generalizations are 'evident' with no proof. Given the paper's scope, that's fine as a remark, but it shouldn't be read as a theorem.

The central characterization (Cor 3.13) does not depend on the black box, so the spine of the paper is solid. The citation pattern is fine; the self-citations are to previous work that is directly relevant, not padding.

Who it's for: anyone working on relative monads or relative pseudomonads, and to a lesser extent semantics people using relative monads. It deserves a proper peer review. My recommendation: send it out, with a request that the authors spell out the dependency in 3.18 and either prove or properly state the enrichment generalization.

Referee Report

2 major / 3 minor

Summary. This paper studies idempotence for relative monads. It introduces two notions: idempotence, meaning invertibility of the extension operator, and the stronger algebraic idempotence, meaning that every algebra for the relative monad is idempotent. The paper exhibits a counterexample showing that the two notions differ, proves that algebraic idempotence is characterized by the forgetful functor from the category of algebras being j-reflective (Corollary 3.13), and gives sufficient conditions under which idempotence implies algebraic idempotence. As a special case, it claims that for relative monads over free cocompletions the two notions coincide (Corollary 3.18). The paper is clearly structured and the main characterization is supported by a sequence of lemmas, though one foundational lemma on equivalent formulations of idempotence appears to be incorrect as stated.

Significance. If the central results stand, the paper provides a clean and useful characterization of algebraic idempotence for relative monads and clarifies a genuine bifurcation that is absent in the non-relative setting. The counterexample in Example 2.9 is valuable, and the characterization in Corollary 3.13 is a natural and apparently correct analogue of the classical fact that a monad is idempotent iff its Eilenberg-Moore forgetful functor is fully faithful. The paper also draws informative connections to prior work of Diers and to relative pseudomonads. However, the erroneous equivalence claims in Lemmas 1.7 and 2.4 need to be corrected, and the advertised free-cocompletion result depends on an unstated external theorem. The paper is likely salvageable, but it is not ready in its present form.

major comments (2)

[Lemmas 1.7 and 2.4] The equivalence claims in Lemma 2.4 are false as stated. Items (3) and (3') assert that E(eta,e) composed with the extension operator equals the identity, which is exactly the first axiom for a T-algebra and therefore holds for every algebra, not only idempotent ones. A split monomorphism in Set need not be an isomorphism, so the proof's inference from the unit law to the equivalence of (1)--(4) is invalid. Concretely, in Example 2.9 the algebra structure on e sending the unique morphism j -> e to f satisfies E(eta,e) \circ \rtimes = id but is not invertible. Since Lemma 1.7 is proved by deferring to Lemma 2.4, the same problem affects the claimed equivalences for idempotent relative monads, including the assertion that condition (3) there is equivalent to invertibility. The authors should correct the lemma, for instance by replacing (3) with the condition that the other composite \rtimes \circ E(eta,e) is the identity, or by adding the hypothesis that E(eta,e) is a monomorphism. This is a load-bearing definitional claim, even though the main characterization in Corollary 3.13 may survive with a corrected lemma.
[Corollary 3.18] The proof of Corollary 3.18, which is advertised in the abstract, relies entirely on [ASS25, Theorem 8.3] as a black box. The hypotheses of that theorem are not stated or verified in the present paper, and the conclusion is used essentially for the equivalence between idempotence and algebraic idempotence for free cocompletion roots. Please reproduce the statement of [ASS25, Theorem 8.3], or at least state its precise hypotheses, and confirm that they hold for every class \Phi of small categories. If the external theorem carries unstated size, pointwise, or smallness conditions, the chain of equivalences in Corollary 3.18 may fail for some classes of colimits. As written, the reader cannot check this dependency.

minor comments (3)

[Setting paragraph, page 1] The claim that the theory generalizes routinely to enriched relative monads and to relative monads in a virtual equipment is asserted without proof or reference. If this is intended as a remark, it would be preferable to label it as an expectation or to provide a reference; as written it reads as an unsupported mathematical claim.
[Proposition 3.12] The terminality assertion is only sketched: the proof says it 'goes through in essentially the same way as in the classical case' and does not spell out the comparison argument. Since terminality is a separate claim from the j-reflectivity used in Corollary 3.13, please expand this part of the proof or explicitly cite a reference where the argument is given in full.
[Proposition 3.8] There is a typo: 'an relative monad' should be 'a relative monad'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's central characterizations are proved in-text, and its cited extension theorem is a normal external dependency rather than a construction-level equivalence.

full rationale

The paper's central chain is internally argued rather than imported from its conclusions. Algebraic idempotence is defined independently in Definition 2.7 as 'every T-algebra is idempotent', and the key characterization Corollary 3.13 is derived from Lemma 3.11 and Proposition 3.12, both proved in the text. Lemma 3.11 establishes that idempotence of a T-algebra is equivalent to every morphism from another algebra carrier being an algebra morphism, and Proposition 3.12 builds the reflective resolution from idempotent algebras; no step in this derivation assumes Corollary 3.13 itself. The weaker notion of idempotence is likewise defined directly via invertibility of the extension operator (Definition 1.3), and the bifurcation is demonstrated by a concrete counterexample (Example 2.9). The only externally dependent step is Corollary 3.18, which invokes [ASS25, Theorem 8.3] to assert that a phi_A-relative monad extends along the free cocompletion to a phi_A-ary monad. This is a citation to prior work by overlapping authors, but the present paper does not redefine or derive the advertised result from that theorem by construction; the theorem supplies an independent extension result whose hypotheses are not reproduced. If that theorem were false or carried hidden hypotheses, Corollary 3.18 would fail, but that is a correctness risk or a missing-support concern, not circularity: the paper's own equations do not reduce to one another, there are no fitted parameters being renamed as predictions, and no uniqueness claim is imported to force the chosen notion. The Setting paragraph's claim that enriched and virtual-equipment generalizations are 'evident' is an unproved scope assertion, but it does not function as a premise in the derivation of the main categorical equivalences. Overall, the core results are self-contained relative to standard background, and the cited dependencies are normal mathematical imports rather than circular inputs.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters or postulated entities appear. The paper introduces a new definition, algebraic idempotence, which is a mathematical concept rather than an invented entity. The central results rest on standard background and on one external theorem from the authors' own prior work.

assumptions (3)

standard math Background theory of relative monads, algebras, Kleisli categories and relative adjunctions from [AM24, ACU15].
Invoked throughout; the paper does not reprove these foundational facts.
domain assumption Every phi_A-relative monad extends along the free cocompletion phi_A to a phi_A-ary monad ([ASS25, Theorem 8.3]).
Load-bearing for Corollary 3.18; cited from the authors' prior work and not proven here.
ad hoc to paper The theory generalises routinely to enriched relative monads and to relative monads in a virtual equipment.
Asserted in the Setting paragraph without proof, and used to claim the results are not special to ordinary categories.

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Cite this review

Pith. "Pith review of Idempotence for relative monads." pith.science (2026). https://pith.science/paper/SAAOSVZT

@misc{pith2026250817794,
  author       = {Pith},
  title        = {Pith review of: Idempotence for relative monads},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SAAOSVZT}},
  note         = {Machine review of arXiv:2508.17794}
}

read the original abstract

We study the concept of idempotence for relative monads, which exhibits several subtleties not present for non-relative monads. In particular, there is a bifurcation of notions of idempotence in the relative setting, which are indistinguishable for idempotent monads. As a special case, we obtain several characterisations of idempotence for monads in extension form.

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Works this paper leans on

2 extracted references · 1 canonical work pages

[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, pages 297–311 (cited on page 1). [ACU15] Thorsten Altenkirch, James Chapman and Tarmo Uustalu. ‘Monads need not be endofunctors’. In:Logical Methods in...

work page 2015
[2025]

[Die75] Yves Diers

arXiv: 2501.12510 (cited on pages 1, 3, 5, 9). [Die75] Yves Diers. ‘J-monades’. In:Comptes Rendus de l’Académie des Sciences280 (1975), pages 1349–1352 (cited on pages 1, 3, 5–7). [FGHW18] Marcelo Fiore, Nicola Gambino, Martin Hyland and Glynn Winskel. ‘Relative pseudo- monads, Kleisli bicategories, and substitution monoidal structures’. In:Selecta Math- ...

arXiv 1975

Pith tools

Reviewed August 15, 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, pages 297–311 (cited on page 1). [ACU15] Thorsten Altenkirch, James Chapman and Tarmo Uustalu. ‘Monads need not be endofunctors’. In:Logical Methods in...

work page 2015

[2] [2025]

[Die75] Yves Diers

arXiv: 2501.12510 (cited on pages 1, 3, 5, 9). [Die75] Yves Diers. ‘J-monades’. In:Comptes Rendus de l’Académie des Sciences280 (1975), pages 1349–1352 (cited on pages 1, 3, 5–7). [FGHW18] Marcelo Fiore, Nicola Gambino, Martin Hyland and Glynn Winskel. ‘Relative pseudo- monads, Kleisli bicategories, and substitution monoidal structures’. In:Selecta Math- ...

arXiv 1975