Homotopy models for $L_{\infty}[1]$-algebras in higher degrees

Taesu Kim

A homotopy model for L_∞[1]-morphisms generalizes A_∞-homotopies and proves a filling condition for simplices with quasi-isomorphism vertices.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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T0 review · grok-4.3

2026-06-30 08:23 UTC pith:CKIBBMD6

load-bearing objection The paper generalizes FOOO A∞-homotopies to L∞[1]-morphisms and claims a filling condition, but lacks visible construction details. the 1 major comments →

arxiv 2606.28985 v1 pith:CKIBBMD6 submitted 2026-06-27 math.AT math.QAmath.SG

Homotopy models for L_(infty)[1]-algebras in higher degrees

Taesu Kim This is my paper

classification math.AT math.QAmath.SG

keywords L_infty[1]-algebrasA_infty-homotopieshigher homotopy theoryquasi-isomorphismssimplicial fillinghomotopy modelsalgebraic topology

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proposes a model of higher homotopy theory for L_∞[1]-morphisms that extends the A_∞-homotopies of Fukaya-Oh-Ohta-Ono. This model treats higher-degree homotopies in a simplicial setting. The authors prove that the model satisfies a filling condition whenever the vertices of a simplex are assigned quasi-isomorphisms. A sympathetic reader would care because the construction supplies a uniform way to build higher homotopies once the vertices satisfy the quasi-isomorphism condition.

Core claim

We propose a model of higher homotopy theory of L_∞[1]-morphisms as a natural generalization of the A_∞-homotopies defined by Fukaya-Oh-Ohta-Ono. Within this framework, we show that a filling condition holds for simplices whose vertices are assigned quasi-isomorphisms.

What carries the argument

The proposed simplicial homotopy model for L_∞[1]-morphisms, which extends A_∞-homotopies to higher degrees and enforces the filling condition on quasi-isomorphism vertices.

Load-bearing premise

The proposed model correctly extends the A_∞-homotopy framework to L_∞[1]-morphisms so that the filling condition for simplices with quasi-isomorphism vertices is both well-defined and provable.

What would settle it

An explicit simplex whose vertices are quasi-isomorphisms but which admits no filling map under the proposed model.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Higher homotopies between L_∞[1]-morphisms can be constructed simplicially once the vertices satisfy the quasi-isomorphism condition.
The filling property supplies a consistent extension of lower-degree A_∞-homotopies to the L_∞[1] setting.
Quasi-isomorphisms between L_∞[1]-algebras admit systematic lifts to higher simplices.
The model organizes the homotopy theory of these morphisms into a simplicial framework.

Where Pith is reading between the lines

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

The same filling mechanism might apply directly to unshifted L_∞-algebras if the model can be adapted without the [1]-shift.
Concrete computations in low-dimensional Lie algebra examples could test whether the filling condition holds in practice.
The construction may link to existing simplicial models used in deformation theory or operadic homotopy.
If the model works, it suggests a uniform simplicial treatment could exist for other shifted algebraic structures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

The paper generalizes FOOO A∞-homotopies to L∞[1]-morphisms and claims a filling condition, but lacks visible construction details.

read the letter

The key point is that this paper puts forward a model for the higher homotopy theory of L_∞[1]-morphisms, generalizing the A_∞-homotopies from Fukaya-Oh-Ohta-Ono, and shows that a filling condition holds when the simplex vertices are quasi-isomorphisms.

What is new is the model and the filling result. It takes the established A_∞ setup and extends it to the L_∞[1] case. This fits into the ongoing work on homotopy algebras, which has applications in deformation theory. The abstract presents it as a natural step.

The paper does well by being direct about the generalization and linking it to the cited reference. It stays focused on the specific claim without unnecessary breadth.

The soft spot is the absence of any actual definitions or proof steps. With only the abstract available, there is no way to check how the model is constructed or whether the filling argument goes through without issues. This leaves the soundness unverified for now.

This work is for people already deep in algebraic topology and infinity-algebra structures. A reader who knows the FOOO paper and works with L_∞[1] things would see the value if the details hold. It is not something that would draw in outsiders.

I would recommend sending it for peer review. The topic is specialized, but if the full paper delivers on the claim with proper constructions, it could be a solid addition to the literature and worth the referees' time.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes a model of higher homotopy theory of L_∞[1]-morphisms as a natural generalization of the A_∞-homotopies defined by Fukaya-Oh-Ohta-Ono, and claims to show that a filling condition holds for simplices whose vertices are assigned quasi-isomorphisms.

Significance. If the proposed model and filling condition can be substantiated with explicit constructions and proofs, the work would extend existing A_∞ homotopy frameworks to the L_∞[1] setting in a manner potentially relevant to higher categorical structures in algebraic topology and deformation theory. No machine-checked proofs, reproducible code, or parameter-free derivations are mentioned.

major comments (1)

The manuscript consists only of the abstract; no definitions of the proposed model, no explicit construction of the homotopy theory, and no proof of the filling condition are supplied. This renders the central claims unverifiable.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The full manuscript on arXiv:2606.28985 contains the definitions, explicit constructions, and proof of the filling condition referenced in the abstract. We address the major comment below.

read point-by-point responses

Referee: The manuscript consists only of the abstract; no definitions of the proposed model, no explicit construction of the homotopy theory, and no proof of the filling condition are supplied. This renders the central claims unverifiable.

Authors: The complete manuscript, including the model for higher homotopy theory of L_∞[1]-morphisms (generalizing FOOO A_∞-homotopies), the explicit constructions, and the proof that the filling condition holds for simplices with quasi-isomorphism vertices, is available in the full text on arXiv:2606.28985. The abstract was provided for reference in this review context, but the arXiv version supplies all required details. We are prepared to excerpt specific sections or answer targeted questions about the constructions if the referee has not yet accessed the full paper. revision: no

Circularity Check

0 steps flagged

No significant circularity; proposal and claim are independent of inputs

full rationale

The abstract proposes a homotopy model for L_∞[1]-morphisms as a generalization of FOOO A_∞-homotopies (external citation to different authors) and states that a filling condition is shown within the framework. No equations, definitions, or self-citations appear that would reduce the claimed result to a fit, renaming, or self-referential input by construction. The derivation chain is not supplied in a form that permits reduction, so the paper is treated as self-contained against external benchmarks with no load-bearing circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review based on abstract only; no free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.1-grok · 5577 in / 1142 out tokens · 47441 ms · 2026-06-30T08:23:44.552727+00:00 · methodology

0 comments

read the original abstract

We propose a model of higher homotopy theory of $L_{\infty}[1]$-morphisms as a natural generalization of the $A_{\infty}$-homotopies defined by Fukaya-Oh-Ohta-Ono \cite{FOOO1}. Within this framework, we show that a filling condition holds for simplices whose vertices are assigned quasi-isomorphisms.

discussion (0)

Forward citations

Cited by 2 Pith papers

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

Kuranishi chart categories and higher cocycle conditions
math.SG 2026-07 unverdicted novelty 7.0

Kuranishi chart categories satisfy a higher homotopical bundle-component cocycle condition automatically, replacing rigid conditions with flexible homotopy-theoretic compatibility.
Categorical structures of Kuranishi spaces with $L_{\infty}[1]$-algebras
math.SG 2026-07 unverdicted novelty 5.0

Defines L∞-Kuranishi spaces via L∞[1]-algebras on Kuranishi charts and proves they form a category embedding smooth manifolds, by modifying conditions from prior work.

Reference graph

Works this paper leans on

8 extracted references · cited by 2 Pith papers

[1]

Non- commut

Lino Amorim, Junwu Tu,The inverse function theorem for curved L-infinity spaces,J. Non- commut. Geom. 16 (2022), no. 4, pp. 1445–1477

2022
[2]

Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono,Lagrangian Intersection Floer Theory : Anomaly and Obstruction Part I, II, 2009

2009
[3]

Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono,Kuranishi structures and Virtual fundamental chain, Springer Monographs in Mathematics, Springer, 2020

2020
[4]

3 (1) 1 - 35, 2001

Bernhard Keller.Introduction to A-infinity algebras and modules,Homology Homotopy Appl. 3 (1) 1 - 35, 2001

2001
[5]

Taesu Kim,Categorical structures for Kuranishi spaces withL ∞[1]-algebras,arXiv: Preprint
[6]

Taesu Kim,Kuranishi internal categories and higher cocycle conditions,arXiv: Preprint
[7]

Alain Prout´ e, Alg´ ebres diff´ erentielles fortement homotopiquement associatives, Th´ ese d’Etat, Universit´ e Paris VII, 1984

1984
[8]

Weibel,An Introduction to Homological Algebra,Cambridge University Press, 1994

Charles A. Weibel,An Introduction to Homological Algebra,Cambridge University Press, 1994

1994

[1] [1]

Non- commut

Lino Amorim, Junwu Tu,The inverse function theorem for curved L-infinity spaces,J. Non- commut. Geom. 16 (2022), no. 4, pp. 1445–1477

2022

[2] [2]

Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono,Lagrangian Intersection Floer Theory : Anomaly and Obstruction Part I, II, 2009

2009

[3] [3]

Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono,Kuranishi structures and Virtual fundamental chain, Springer Monographs in Mathematics, Springer, 2020

2020

[4] [4]

3 (1) 1 - 35, 2001

Bernhard Keller.Introduction to A-infinity algebras and modules,Homology Homotopy Appl. 3 (1) 1 - 35, 2001

2001

[5] [5]

Taesu Kim,Categorical structures for Kuranishi spaces withL ∞[1]-algebras,arXiv: Preprint

[6] [6]

Taesu Kim,Kuranishi internal categories and higher cocycle conditions,arXiv: Preprint

[7] [7]

Alain Prout´ e, Alg´ ebres diff´ erentielles fortement homotopiquement associatives, Th´ ese d’Etat, Universit´ e Paris VII, 1984

1984

[8] [8]

Weibel,An Introduction to Homological Algebra,Cambridge University Press, 1994

Charles A. Weibel,An Introduction to Homological Algebra,Cambridge University Press, 1994

1994