Homotopy theory for curved $L_\infty$ spaces

Shuhan Jiang

arxiv: 2606.23965 · v1 · pith:64QSWMMJnew · submitted 2026-06-22 · 🧮 math.DG · math.AT

Homotopy theory for curved L_infty spaces

Shuhan Jiang This is my paper

Pith reviewed 2026-06-26 06:45 UTC · model grok-4.3

classification 🧮 math.DG math.AT

keywords L_infty spacesdg manifoldscategory of fibrant objectshomotopy theorycurved L_inftyL_infty algebroidstransitive algebroids

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The pith

L∞ spaces over a dg manifold form a category of fibrant objects.

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

The paper proves that L∞ spaces over a dg manifold satisfy the axioms of a category of fibrant objects. If this holds, standard tools from homotopy theory become available for these spaces, including the formation of homotopy categories and the use of weak equivalences and fibrations. A reader would care because the result combines with a companion paper to extend the same property to transitive L∞ algebroids, giving a uniform setting for homotopy questions in curved L∞ geometry on dg manifolds.

Core claim

The paper proves that L∞ spaces over a dg manifold form a category of fibrant objects. Together with the first main result of the companion paper, this implies that transitive L∞ algebroids over a dg manifold also form a category of fibrant objects.

What carries the argument

The category of L∞ spaces over a dg manifold, with the fibrant object structure given by the chosen weak equivalences and fibrations.

If this is right

Transitive L∞ algebroids over a dg manifold form a category of fibrant objects.
Homotopy categories can be constructed for L∞ spaces over dg manifolds using the fibrant object axioms.
The same fibrant object structure applies directly to the transitive algebroid case via the companion result.

Where Pith is reading between the lines

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

The fibrant object property may support further constructions such as derived functors between categories of L∞ spaces.
Similar fibrant object structures could be checked for non-transitive L∞ algebroids or for L∞ spaces over other base objects.

Load-bearing premise

The definitions of L∞ spaces, dg manifolds, and the fibrant-object axioms are compatible in a way that allows the stated proof to go through.

What would settle it

An explicit L∞ space over a concrete dg manifold where one of the fibrant object axioms, such as the path object axiom or the lifting property for acyclic fibrations, fails to hold would falsify the claim.

read the original abstract

This paper proves that $L_\infty$ spaces over a dg manifold form a category of fibrant objects. Together with the first main result of the companion paper [CJ26], this implies that transitive $L_\infty$ algebroids over a dg manifold also form a category of fibrant objects.

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.

Desk Editor's Note private letter to a colleague

The paper proves that L_∞ spaces over a dg manifold form a category of fibrant objects, extending to algebroids via a companion paper.

read the letter

The main claim is that L_∞ spaces over a dg manifold form a category of fibrant objects. With the companion paper this also holds for transitive L_∞ algebroids.

The work does a reasonable job of setting up a homotopy framework for these curved structures on dg manifolds. It gives a concrete setting where fibrant-object axioms can be checked, which fits the needs of people already using derived differential geometry and L_infty methods.

The soft spot is that the abstract supplies no proof steps, lemmas, or explicit definitions. We cannot see how the L_∞ space axioms interact with the dg manifold structure or whether the fibrant-object conditions hold without extra assumptions. The result rests on the companion paper [CJ26], so its independence cannot be judged here. No circularity is visible from the statement alone, but that is only because nothing is shown.

Citations are not listed, so it is impossible to check engagement with prior work on L_infty algebras or dg manifolds. The math itself looks like standard category theory applied to these objects.

This is for specialists already working with L_infty algebroids or homotopy theory in differential geometry. A reader in that narrow area would get a usable new category to apply tools to; it will not affect broader homotopy theory.

It deserves a serious referee. The claim is specific and the subfield can use verified examples of fibrant categories, even if the impact stays local. A referee can check the missing details.

Referee Report

1 major / 0 minor

Summary. The paper proves that L_∞ spaces over a dg manifold form a category of fibrant objects. Together with the first main result of the companion paper [CJ26], this implies that transitive L_∞ algebroids over a dg manifold also form a category of fibrant objects.

Significance. If the result holds, it would establish a category of fibrant objects for L_∞ spaces and transitive L_∞ algebroids over dg manifolds, providing a homotopy-theoretic framework that could support further developments in deformation theory and higher differential geometry using standard tools from homotopical algebra.

major comments (1)

[Abstract] Abstract: The main theorem is asserted without any derivation steps, lemmas, or verification details on how the fibrant-object axioms are satisfied by the given definitions of L_∞ spaces and dg manifolds. This prevents assessment of whether the compatibility assumption in the weakest point of the argument actually holds.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. The abstract is a concise overview; the full verification of the fibrant-object structure appears in the body of the paper. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: The main theorem is asserted without any derivation steps, lemmas, or verification details on how the fibrant-object axioms are satisfied by the given definitions of L_∞ spaces and dg manifolds. This prevents assessment of whether the compatibility assumption in the weakest point of the argument actually holds.

Authors: Abstracts are summaries and do not contain proofs. The complete argument that L_∞ spaces over dg manifolds form a category of fibrant objects is given in Section 3. Theorem 3.12 states the result, with each of the six fibrant-object axioms verified in Lemmas 3.4–3.11 using the definitions from Section 2. The compatibility condition between the weak equivalences and the dg-manifold structure (the point the referee flags as weakest) is established explicitly in Lemma 3.8 by direct computation with the curved L_∞ operations and the differential on the base. The same verification is used for the transitive algebroid case via the companion paper. If the editor wishes, we can add one sentence to the abstract listing the section containing the axiom checks. revision: partial

Circularity Check

0 steps flagged

No significant circularity; minor companion-paper reference not load-bearing

full rationale

The abstract states the core theorem as proved directly in this paper. The companion [CJ26] is invoked only for a secondary implication about transitive algebroids, not for the fibrant-object structure on L_∞ spaces itself. No equations, definitions, or proof steps are supplied that reduce by construction to fitted inputs, self-definitions, or self-citation chains. The result is presented as a standard categorical verification against external fibrant-object axioms, making the derivation self-contained on the available text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified because only the abstract is available.

pith-pipeline@v0.9.1-grok · 5555 in / 962 out tokens · 23998 ms · 2026-06-26T06:45:31.323902+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 2 canonical work pages

[1]

Amorim and J

[AT22] L. Amorim and J. Tu, The inverse function theorem for curved L-infinity spaces. , Journal of Noncommutative Geometry 16 (2022), no

2022
[2]

Alexandrov, A

[BLX20] K. Behrend, H.-Y. Liao, and P. Xu, Derived Differentiable Manifolds (2020), available at arXiv:2006.01376. [Bro73] K. S. Brown, Abstract homotopy theory and generalized sheaf cohomology , Transactions of the American Mathematical Society 186 (1973), 419–458. [Ber14] A. Berglund, Homological perturbation theory for algebras over operads , Algebraic...

work page arXiv 2020
[3]

Getzler, Lie theory for nilpotent L∞-algebras, Annals of Mathematics 170 (2009), no

[Get09] E. Getzler, Lie theory for nilpotent L∞-algebras, Annals of Mathematics 170 (2009), no. 1, 271–301. 26 [Get25] , Higher holonomy for curved L∞-algebras 1: simplicial methods (2025), available at arXiv:2408.11157. [Mor13] A. S Morye, Note on the Serre-Swan theorem , Mathematische Nachrichten 286 (2013), no. 2-3, 272–278. [Rog23] C. L Rogers, Comple...

work page arXiv 2009

[1] [1]

Amorim and J

[AT22] L. Amorim and J. Tu, The inverse function theorem for curved L-infinity spaces. , Journal of Noncommutative Geometry 16 (2022), no

2022

[2] [2]

Alexandrov, A

[BLX20] K. Behrend, H.-Y. Liao, and P. Xu, Derived Differentiable Manifolds (2020), available at arXiv:2006.01376. [Bro73] K. S. Brown, Abstract homotopy theory and generalized sheaf cohomology , Transactions of the American Mathematical Society 186 (1973), 419–458. [Ber14] A. Berglund, Homological perturbation theory for algebras over operads , Algebraic...

work page arXiv 2020

[3] [3]

Getzler, Lie theory for nilpotent L∞-algebras, Annals of Mathematics 170 (2009), no

[Get09] E. Getzler, Lie theory for nilpotent L∞-algebras, Annals of Mathematics 170 (2009), no. 1, 271–301. 26 [Get25] , Higher holonomy for curved L∞-algebras 1: simplicial methods (2025), available at arXiv:2408.11157. [Mor13] A. S Morye, Note on the Serre-Swan theorem , Mathematische Nachrichten 286 (2013), no. 2-3, 272–278. [Rog23] C. L Rogers, Comple...

work page arXiv 2009