Day convolution for algebraic patterns

F\'elix Loubaton; Jaco Ruit; Thomas Blom

REVIEW 1 major objections 3 minor 12 references

Day convolution for algebraic patterns

T0 review · 1 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper proves that for robust algebraic patterns, an algebrad is exponentiable if and only if it satisfies the Conduché criterion (CC).

desk verdict A serious paper in which the advertised necessity theorem for equivariant operads currently rests on a false pullback-preservation claim in Example 7.24. read the letter →

arxiv 2603.29815 v2 submitted 2026-03-31 math.CT math.AT

Thomas Blom , Félix Loubaton , Jaco Ruit This is my paper · ORCID

classification math.CTmath.AT MSC 18N6018N70

keywords algebraicpatternsweakSegalfibrationsalgebradsDayconvolutionexponentiableobjectsConduchécriterion∞-operadsvirtualdoublecategories

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 asks when Day convolution exists for operad-like structures beyond ∞-operads. Working in the framework of algebraic patterns, where such structures are packaged as weak Segal fibrations (renamed algebrads), it proves that an algebrad over a pattern O is exponentiable precisely when a Conduché-style condition holds: for every composable pair of active maps ending at an elementary object, the ∞-category of factorizations of a lift is weakly contractible. The sufficiency direction works for every algebraic pattern, while the necessity direction is proved under an extra robustness package of hypotheses that the standard examples satisfy. If the characterization is correct, Day convolution is available inside the categories of (equivariant) ∞-operads and virtual double ∞-categories, and the underlying structures of exponential objects can be computed from the underlying structures of their factors.

What carries the argument

The load-bearing mechanism is the tree ∞-category Ω[O] of an algebraic pattern O: its objects are strings t0⇝⋯⇝tn of active morphisms such that tn is elementary, and Theorem D presents O-algebrads as complete Segal presheaves on Ω[O]. This turns exponentiability into a question about whether a right adjoint between presheaf categories preserves complete Segal objects, which can be handled by simplicial combinatorics. The Conduché criterion (CC) itself asks that the factorization ∞-category Fact(f|g∘h) over active composites ending at an elementary object be weakly contractible. Robustness—a package of soundness, saturation, inert-map detection, and finiteness conditions on the π0 functor—supplies the partial-composite (grafting) construction used to prove the necessity direction.

What would settle it

Find an exponentiable algebrad over a robust algebraic pattern for which some factorization ∞-category Fact(f|g∘h) over an active composite ending at an elementary object is not weakly contractible; that would refute Theorem B. For the non-robust non-symmetric pattern Δ^op,♭, an exponentiable non-symmetric ∞-operad violating condition (CC) would show robustness is genuinely needed and would delimit the theorem's scope.

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Extended reading notes

Core claim

The central claim is that exponentiable objects in the ∞-category Algad(O) of O-algebrads are exactly the maps P→O satisfying condition (CC): for any composable active pair x⇝y⇝e with e elementary and any lift f of the composite, the factorization ∞-category Fact(f|g∘h) is weakly contractible. Theorem A establishes sufficiency for all algebraic patterns; Theorem B establishes necessity for robust algebraic patterns, giving complete characterizations for ∞-operads, equivariant ∞-operads, and virtual double ∞-categories. In the same framework, Theorem D identifies Algad(O) with the ∞-category of complete Segal presheaves on a tree category Ω[O] whose objects are strings of active morphisms ending in an elementary object, and Theorem C shows that the underlying-graph functor Γ preserves exponential objects: Γ[P,Q] is the internal hom [ΓP,ΓQ].

Load-bearing premise

The necessity direction (Theorem B) rests on the robustness conditions of Definition 7.10; if a pattern is not robust, the paper does not establish that the Conduché criterion is necessary, and the non-symmetric operad pattern is explicitly left uncovered.

Editorial extensions

If this is right

For ∞-operads, equivariant ∞-operads, and virtual double ∞-categories, exponentiable objects are completely characterized by condition (CC); for ∞-operads this recovers the flatness criterion of Hinich.
If P is exponentiable, the underlying graph of the exponential object [P,Q] is the internal hom [ΓP,ΓQ], generalizing the familiar fact that the underlying ∞-category of a Day convolution is a functor category.
Every algebraic pattern O admits the equivalence Algad(O)≃CSeg(Ω[O]), and iterating the tree construction produces a tower of inclusions Algad(O)→Algad(Ω[O]^op)→⋯ whose images consist of exponentiable objects.
Every Segal O-category, viewed as an O-algebrad, is exponentiable; in particular, every double category is exponentiable as a virtual double category.
For any ∞-category C, the virtual cospan double category Cospan^virt(C) is an exponentiable virtual double category, even when C lacks pushouts.

Reading between the lines

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

A natural next step this opens up is to use the exponentials constructed here to define presheaf algebrads and develop Yoneda, Kan extension, and cocompletion technology for generalized operads and virtual double categories; the paper indicates such a sequel but leaves the development implicit.
Because the non-symmetric operad pattern Δ^op,♭ is explicitly not robust, the necessity direction for non-symmetric ∞-operads remains open; a natural test is whether another robust replacement or a sharpened robustness condition can cover it.
The tree-category presentation suggests that any algebraic pattern admits Segal-space models with the same formal behavior as Rezk's complete Segal spaces, which may let the exponentiability criterion be checked purely combinatorially in examples beyond those listed.

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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 substantial paper, but the necessity theorem for equivariant operads is not currently proven as written. The stress-test concern is correct. Example 7.24 asserts that the orbit functor O:F_G→F commutes with pullbacks. It does not. For G=C2, take X=Y to be the regular two-element G-set and Z the trivial two-element G-set. Let f,g:X,Y→Z be constant at a fixed point of Z. In G-Set, X×_Z Y is X×Y with the diagonal action and has two orbits, while O(X)=O(Y)=* and O(Z) has two points, so the set-theoretic pullback O(X)×_{O(Z)}O(Y) is a point. Since composition in Span(F_G) is defined by pullback, the claimed pattern map Span(F_G)^♭→Span(F)^♭ is not obtained by the stated argument, and condition (4b) in Definition 7.10 is unverified. Consequently Example 8.25 does not establish Theorem B for G-∞-operads. This is load-bearing, because necessity for equivariant operads is one of the paper's headline claims.

The rest of the paper has real value. Theorem D, the description of algebrads as complete Segal presheaves on the tree category, is a genuine and useful new formulation. Theorem A cleanly unifies known sufficiency results, and Theorem C on underlying graphs is a nice observation. The virtual double category example, Δ^op,♮, is robust by a different and more convincing argument, so the flaw seems specific to the equivariant G-set example rather than to the general framework. The proof architecture is coherent overall, although many ∞-categorical details are delegated to external sources and I could not fully verify all of them. The forward reference from Proposition 4.22 to Corollary 8.6 is awkward, but the parenthetical alternative proof suggests it is not a real circularity.

This paper deserves a serious referee, but the referee should ask for a corrected or replaced robustness proof for Span(F_G)^♭ before the equivariant necessity claim is accepted. If that gets fixed, this will be a strong contribution to the algebraic patterns literature.

Referee Report

1 major / 3 minor

Summary. The paper develops a general theory of Day convolution for algebraic patterns by characterizing exponentiable objects (algebrads) over a pattern O. Its main results are: Theorem A gives a sufficient Conduché-type criterion (CC) for exponentiability of an object in Algad(O); Theorem B asserts that for robust patterns the criterion is also necessary; Theorem D describes Algad(O) as complete Segal presheaves on a tree category Ω[O]; and Theorem C states that the underlying-graph functor preserves exponential objects. The paper also works out the criterion for ∞-operads, equivariant ∞-operads, generalized operads, and virtual double ∞-categories, and it gives an explicit counterexample showing that the sufficient criterion is strictly weaker than exponentiability of the underlying categorical functor on active morphisms.

Significance. If correct, the paper would substantially advance the subject: it provides the first necessary-and-sufficient Conduché criterion for exponentiability in several important ∞-categorical contexts, establishes a tree-category model for algebrads, and proves compatibility of exponential objects with underlying graphs. The proof strategy, passing through complete Segal presheaves on Ω[O], is genuinely different from the Lurie–Hinich–Nardin–Shah approach and is clearly explained. The paper also gives concrete worked examples and an explicit counterexample showing that the general criterion is sharp. However, one load-bearing example needed for the equivariant conclusion is not proved correctly as written, and this affects the advertised necessity theorem for equivariant ∞-operads.

major comments (1)

[§7.4, Example 7.24 and §8.5, Example 8.25] Example 7.24 asserts that the orbit-set functor O: F_G → F, sending a finite G-set to its set of orbits, commutes with pullbacks, and uses this to conclude that the functor Span(F_G)^♭ → Span(F)^♭ is the pattern map π0 and that condition (4b) of Definition 7.10 holds. This pullback-preservation claim is false in general. For G = C2, let X and Y be the two-element regular G-set and let Z be the two-element G-set with trivial action. Let f,g: X,Y → Z be the constant maps at a fixed point z0. These are G-equivariant maps. In G-Set, the pullback X ×_Z Y is X × Y with the diagonal G-action, whose orbit set has two elements. But O(X) and O(Y) are singletons, while O(Z) has two elements, and the two induced maps O(X) → O(Z) and O(Y) → O(Z) both pick the same fixed point, so O(X) ×_{O(Z)} O(Y) is a singleton. Thus O does not preserve this pullback, and the claimed pattern map Span(F_G)^♭ → Span(F)^♭ is not obtained by the stated argument. Since Example 8.25 invokes exactly this robustness of Span(F_G)^♭ to deduce the necessary direction of the Conduché criterion for G-operads, Theorem B is not established for the equivariant case as written. This is a load-bearing gap in the proof of the paper's headline equivariant application, although it does not by itself invalidate the general framework or the other examples.

minor comments (3)

[§4.3, Proposition 4.22] The proof of Proposition 4.22 invokes Example 7.30 and Corollary 8.6 before those results are introduced. The reference is not circular—Corollary 8.6 is proved independently of Proposition 4.22—but the forward dependence should be stated explicitly and ideally reorganized to avoid the appearance of circularity.
[§7.2, Definition 7.10] In condition (4c), the phrase “if x ∈ O lies over n” is not defined before it is used. It would be clearer to state explicitly that this means under the composed functor O → Span(F)^♭, or to spell out the intended projection to finite sets.
[§9.3, Construction 9.7] The toy example is described mainly through diagrams and picture references. A precise set-theoretic definition of the maps A → B, or at least an explicit description of the corresponding graph maps in Fun(G,Cat), would make the counterexample easier to verify independently.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are derived from new robustness hypotheses and an independent tree-category/presheaf equivalence; self-citations are not load-bearing.

full rationale

The derivation chain is self-contained with respect to the usual circularity hazards. Theorem A is proved by establishing Theorem D (Algad(O) ≃ CSeg(Ω[O])) and then working in the presheaf category; the Conduché-style condition (CC) is a genuine sufficient condition, not a re-encoding of exponentiability. Theorem B introduces robustness (Definition 7.10) as a package of new hypotheses; the grafting proof uses those hypotheses to produce partial composites and does not feed the target statement back into the assumptions. The only self-citation with a load-bearing appearance is [Blo24, Corollary 6.2] in Construction 7.6, but the paper explicitly offers [AF20, Theorem 0.8] as an independent source for the same straightening result, so even that citation is not load-bearing. [Rui25a, Corollary 6.9] and [Rui25b] are used to construct the π0 oplax functor; these are general double-categorical framework results, not the target characterization, and no fitted parameter is renamed as a prediction. Two non-circular caveats should be noted for correctness rather than circularity: Example 7.24's 'one readily verifies that this commutes with pullbacks' for the orbit functor is asserted without proof and is questioned by the C2 counterexample described in the skeptic analysis, so the robustness verification for equivariant operads—and hence Theorem B for that example—is at risk; and Example 7.23 explicitly states that Δ^op,♭ is not robust and gives no robust replacement, so the necessity direction is not claimed for non-symmetric operads. Neither caveat is a circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

There are no free parameters and no invented physical entities. The central theorems rest on background ∞-categorical formalism and on the new technical condition of robustness, which is verified for the main examples and used to prove necessity.

assumptions (6)

standard math Quasi-categorical foundations and straightening/unstraightening from Lurie's HTT and HA.
Throughout the paper, results about ∞-categories, cocartesian fibrations, and left fibrations are imported without proof, for example in Lemma 4.20 and Theorem 4.21.
domain assumption Chu-Haugseng theory of algebraic patterns and weak Segal fibrations, including soundness and saturatedness.
Definitions 2.1 through 2.8 and later uses of soundness and saturation rest on this framework; the paper extends it rather than rederiving it.
domain assumption Juran's equivalence between factorization systems and factorization double categories.
Corollary 2.19 identifies algebraic patterns with double patterns, and all later double-categorical formulations of algebrads depend on this equivalence.
domain assumption Equivalence results between specific patterns, such as Algad(Span(F)^♭) with Op and similar equivalences from BHS25 and CHH18.
Examples 8.23 through 8.25 use these equivalences to extend Theorem B to non-robust patterns such as F_*^♭; if these equivalences fail, those conclusions would not follow.
standard math Ayala-Francis Conduché criterion for exponentiable functors in Cat/∞.
This is the model for condition (CC) and is invoked in Lemma 5.4 and Proposition 9.5.
domain assumption Robustness, Definition 7.10, as a standing assumption for Theorem B.
Soundness, saturation, π0-strongness, and existence of π0-cocartesian inert splittings are used in the grafting construction of Section 8; Theorem B is not claimed outside this class.

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Cite this review

Pith. "Pith review of Day convolution for algebraic patterns." pith.science (2026). https://pith.science/paper/2603.29815

@misc{pith2026260329815,
  author       = {Pith},
  title        = {Pith review of: Day convolution for algebraic patterns},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2603.29815}},
  note         = {Machine review of arXiv:2603.29815}
}

abstract

We characterize the exponentiable objects for a wide range of structures prevalent in $\infty$-categorical algebra, extending the construction of Day convolution to more general structures than $\infty$-operads. More precisely, we give a criterion that is both necessary and sufficient for many of these structures encountered in practice, such as (equivariant) $\infty$-operads and virtual double $\infty$-categories. We work within the framework of algebraic patterns of Chu-Haugseng that describe these structures in terms of weak Segal fibrations. As part of the proof, we give a new description of weak Segal fibrations in terms of generalized Segal spaces on certain "tree" categories. We also define the "underlying graph" of a weak Segal fibration, extending the notion of the underlying $\infty$-category for $\infty$-operads, and explicitly describe the underlying graph of exponential objects in weak Segal fibrations.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 6 canonical work pages

[1]

[AF20] David Ayala and John Francis,Fibrations of∞-categories, Higher Structures4(2020), no. 1, 168–

work page 2020
[6]

[CH23] ,Free algebras through Day convolution, Algebraic & Geometric Topology22(2023), no

[CH21] Hongyi Chu and Rune Haugseng,Homotopy-coherent algebra via Segal conditions, Advances in Mathematics385(2021), 107733. [CH23] ,Free algebras through Day convolution, Algebraic & Geometric Topology22(2023), no. 7, 3401–3458. [CHH18] Hongyi Chu, Rune Haugseng, and Gijs Heuts,Two models for the homotopy theory of∞-operads, Journal of Topology11(2018),...

work page 2021
[7]

[Jur25] Branko Juran,On orthogonal factorization systems and double categories, arXiv:2501.01363,

arXiv
[8]

[Ker23] David Kern,Monoidal envelopes and Grothendieck construction for dendroidal Segal objects, arXiv:2301.10751,

arXiv
[12]

[Tho25] Ea E Thompson,Pro-representable virtual double categories, talk at the Topos Institute Berkeley Seminar, slides,https://categorytheory.zulipchat.com/user_uploads/21317/ kFFsTvEkBvYe_xZOhIkeem8f/Pro-representableVDCs.pdf, 2025, accessed on 9 Sep

work page 2025
[265]

[Ark25a] Nathanael Arkor,Exponentiable virtual double categories and presheaves for double categories, arXiv:2508.11611,

[AFR18] David Ayala, John Francis, and Nick Rozenblyum,Factorization homology I: Higher categories, Ad- vances in Mathematics333(2018), 1042–1177. [Ark25a] Nathanael Arkor,Exponentiable virtual double categories and presheaves for double categories, arXiv:2508.11611,

arXiv 2018
[541]

3, 973–1007

74 [Rez01] Charles Rezk,A model for the homotopy theory of homotopy theory, Transactions of the American Mathematical Society353(2001), no. 3, 973–1007. [Rui25a] Jaco Ruit,Homotopy coherent companionships and conjunctions, arXiv:2408.14335,

arXiv 2001
[2017]

3, 1441–1470

[MW07a] Ieke Moerdijk and Ittay Weiss,Dendroidal sets, Algebraic & Geometric Topology7(2007), no. 3, 1441–1470. [MW07b] ,Dendroidal sets, Algebr. Geom. Topol.7(2007), 1441–1470. [NS22] Denis Nardin and Jay Shah,Parametrized and equivariant higher algebra, arXiv:2203.00072,

arXiv 2007

Show all 12 references

[2022]

Pereira,Equivariant dendroidal sets, Algebraic & Geometric Topology18(2018), no

[Per18] Lu ´ıs A. Pereira,Equivariant dendroidal sets, Algebraic & Geometric Topology18(2018), no. 4, 2179–2244. [Pis14] Claudio Pisani,Sequential multicategories, Theory and Applications of Categories29(2014), 496–

2018
[2024]

Pereira,Equivariant dendroidal segal spaces and G–∞–operads, Alge- braic & Geometric Topology20(2020), no

[BP20] Peter Bonventre and Lu ´ıs A. Pereira,Equivariant dendroidal segal spaces and G–∞–operads, Alge- braic & Geometric Topology20(2020), no. 6, 2687–2778. [BP22] ,Equivariant dendroidal sets and simplicial operads, Journal of Topology15(2022), no. 2, 745–805. [BS25] Shaul B...

2020
[2025]

co/files/Three-dimensional%20structures%20(CT%202025).pdf, 2025, accessed on 12 Feb

73 [Ark25b] ,The three-dimensional structures formed by monoidal categories, bicategories, double cat- egories, etc., talk at the International Category Theory Conference, slides,https://arkor. co/files/Three-dimensional%20structures%20(CT%202025).pdf, 2025, accessed on 12 Feb

2025
[2026]

4, 1893–1959

[Bar18] Clark Barwick,From operator categories to higher operads, Geometry & Topology22(2018), no. 4, 1893–1959. [BHS25] Shaul Barkan, Rune Haugseng, and Jan Steinebrunner,Envelopes for algebraic patterns, Algebraic & Geometric Topology25(2025), no. 9, 5319–5388. [Blo24] Thoma...

2018

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

[AF20] David Ayala and John Francis,Fibrations of∞-categories, Higher Structures4(2020), no. 1, 168–

work page 2020

[2] [6]

[CH23] ,Free algebras through Day convolution, Algebraic & Geometric Topology22(2023), no

[CH21] Hongyi Chu and Rune Haugseng,Homotopy-coherent algebra via Segal conditions, Advances in Mathematics385(2021), 107733. [CH23] ,Free algebras through Day convolution, Algebraic & Geometric Topology22(2023), no. 7, 3401–3458. [CHH18] Hongyi Chu, Rune Haugseng, and Gijs Heuts,Two models for the homotopy theory of∞-operads, Journal of Topology11(2018),...

work page 2021

[3] [7]

[Jur25] Branko Juran,On orthogonal factorization systems and double categories, arXiv:2501.01363,

arXiv

[4] [8]

[Ker23] David Kern,Monoidal envelopes and Grothendieck construction for dendroidal Segal objects, arXiv:2301.10751,

arXiv

[5] [12]

[Tho25] Ea E Thompson,Pro-representable virtual double categories, talk at the Topos Institute Berkeley Seminar, slides,https://categorytheory.zulipchat.com/user_uploads/21317/ kFFsTvEkBvYe_xZOhIkeem8f/Pro-representableVDCs.pdf, 2025, accessed on 9 Sep

work page 2025

[6] [265]

[Ark25a] Nathanael Arkor,Exponentiable virtual double categories and presheaves for double categories, arXiv:2508.11611,

[AFR18] David Ayala, John Francis, and Nick Rozenblyum,Factorization homology I: Higher categories, Ad- vances in Mathematics333(2018), 1042–1177. [Ark25a] Nathanael Arkor,Exponentiable virtual double categories and presheaves for double categories, arXiv:2508.11611,

arXiv 2018

[7] [541]

3, 973–1007

74 [Rez01] Charles Rezk,A model for the homotopy theory of homotopy theory, Transactions of the American Mathematical Society353(2001), no. 3, 973–1007. [Rui25a] Jaco Ruit,Homotopy coherent companionships and conjunctions, arXiv:2408.14335,

arXiv 2001

[8] [2017]

3, 1441–1470

[MW07a] Ieke Moerdijk and Ittay Weiss,Dendroidal sets, Algebraic & Geometric Topology7(2007), no. 3, 1441–1470. [MW07b] ,Dendroidal sets, Algebr. Geom. Topol.7(2007), 1441–1470. [NS22] Denis Nardin and Jay Shah,Parametrized and equivariant higher algebra, arXiv:2203.00072,

arXiv 2007