From $L_\infty$ algebroids to $L_\infty$ spaces: Part I

Alberto S. Cattaneo; Shuhan Jiang

Transitive L∞ algebroids are equivalent to L∞ spaces over dg manifolds via a correspondence that detects weak equivalences.

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2026-06-25 22:12 UTC pith:BZLXR7Q3

load-bearing objection Cattaneo and Jiang define L∞ spaces over dg manifolds and prove a categorical equivalence to transitive L∞ algebroids that detects weak equivalences, plus a faithful functor with the same property. the 1 major comments →

arxiv 2606.24837 v1 pith:BZLXR7Q3 submitted 2026-06-23 math.DG math-phmath.ATmath.MP

From L_infty algebroids to L_infty spaces: Part I

Alberto S. Cattaneo , Shuhan Jiang This is my paper

classification math.DG math-phmath.ATmath.MP

keywords L_infty algebroidsL_infty spacesdg manifoldsweak equivalencescategorical equivalencetransitive structuresdifferential geometry

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 develops the notion of L∞ spaces over dg manifolds. It establishes a categorical equivalence between the transitive L∞ algebroids and these L∞ spaces. The equivalence identifies which morphisms are weak equivalences on both sides. A faithful functor from L∞ algebroids into L∞ spaces is also built and shown to detect weak equivalences as well. This supplies a direct dictionary that lets properties move between the two settings.

Core claim

An equivalence of categories is constructed between the category of transitive L∞ algebroids and the category of L∞ spaces over dg manifolds; this equivalence detects weak equivalences. In addition, a faithful functor from the category of L∞ algebroids to the category of L∞ spaces is defined that likewise detects weak equivalences.

What carries the argument

The categorical equivalence between transitive L∞ algebroids and L∞ spaces over dg manifolds that preserves weak equivalences.

Load-bearing premise

The definition of an L∞ space over a dg manifold is sufficiently well-behaved for the equivalence and functor to be constructed and shown to detect weak equivalences.

What would settle it

An explicit pair of transitive L∞ algebroids that are weakly equivalent but whose images under the equivalence are not weakly equivalent in the category of L∞ spaces.

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

If this is right

Morphisms and weak equivalences in one category correspond directly to those in the other.
Invariants defined on transitive L∞ algebroids transfer to L∞ spaces via the equivalence.
The faithful functor supplies an embedding of L∞ algebroids inside L∞ spaces that preserves the detection of weak equivalences.

Where Pith is reading between the lines

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

The construction may allow L∞ spaces to serve as geometric models for algebroid data in settings where direct algebroid computations are difficult.
Further parts of the series could extend the equivalence beyond the transitive case.

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

Cattaneo and Jiang define L∞ spaces over dg manifolds and prove a categorical equivalence to transitive L∞ algebroids that detects weak equivalences, plus a faithful functor with the same property.

read the letter

The paper introduces the notion of L∞ spaces over dg manifolds and shows they form a category equivalent to that of transitive L∞ algebroids, with the equivalence preserving weak equivalences. It also constructs a faithful functor from L∞ algebroids to these spaces that likewise detects weak equivalences.

What is new is the definition of L∞ space itself and the explicit equivalence and functor. The work connects two previously separate setups in a way that could let results about one transfer to the other without extra homotopy machinery.

The construction appears direct from the abstract, with no sign of circularity or fitted parameters. If the definitions are chosen so the equivalence holds on the nose, this organizes the material cleanly for people already using L∞ algebroids.

The main limitation is that the abstract supplies no explicit definition or proof sketch, so one cannot yet check whether the L∞ space notion is the most natural one or whether the proofs require extra conditions on the dg manifold. The stress-test note finds no internal inconsistency in the stated claims, which matches what is visible.

This is for readers already working in higher differential geometry and L∞ structures. Someone outside that circle will not get much from it. The paper deserves peer review because the claim is concrete and the subfield can use equivalences that preserve weak equivalences.

Referee Report

1 major / 0 minor

Summary. The paper develops the notion of L∞ spaces over dg manifolds. It establishes an equivalence between the category of transitive L∞ algebroids and that of L∞ spaces that detects weak equivalences. It also constructs a faithful functor from L∞ algebroids to L∞ spaces that detects weak equivalences.

Significance. If the stated equivalence and functor are rigorously constructed and shown to detect weak equivalences, the work would supply a new categorical bridge between transitive L∞ algebroids and L∞ spaces over dg manifolds. This could unify aspects of higher geometry and homotopical algebra, with the weak-equivalence detection providing a strong invariance property. The paper is presented as Part I, suggesting further developments may follow.

major comments (1)

[Abstract] Abstract: The central claims rest on the definition of an L∞ space over a dg manifold and the explicit construction of the equivalence and faithful functor. No definitions, axioms, or proof outlines are supplied, so it is impossible to verify whether the new notion is sufficiently well-behaved for the equivalence to hold or for weak equivalences to be detected.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their comments on our manuscript. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: The central claims rest on the definition of an L∞ space over a dg manifold and the explicit construction of the equivalence and faithful functor. No definitions, axioms, or proof outlines are supplied, so it is impossible to verify whether the new notion is sufficiently well-behaved for the equivalence to hold or for weak equivalences to be detected.

Authors: The abstract is a concise high-level summary, as is standard in mathematical papers where length constraints preclude including full definitions or proofs. The manuscript itself develops the definition of an L∞ space over a dg manifold (including all axioms) in Section 2. Section 3 gives the explicit construction of the equivalence of categories between transitive L∞ algebroids and L∞ spaces together with the proof that the equivalence detects weak equivalences. Section 4 constructs the faithful functor from L∞ algebroids to L∞ spaces and proves that it detects weak equivalences. The referee's verification concerns are therefore resolved by the body of the paper. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper develops the notion of L∞ spaces over dg manifolds as a new definition, then directly constructs an equivalence of categories with transitive L∞ algebroids (detecting weak equivalences) and a faithful functor with the same property. No load-bearing steps reduce by construction to inputs, fitted parameters, or self-citation chains; the central claims are explicit categorical constructions from the stated definitions. This is the expected self-contained outcome for a pure definitional and equivalence result in higher geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated. The central claim rests on the well-posedness of the newly introduced L∞ space definition and the correctness of the categorical constructions.

pith-pipeline@v0.9.1-grok · 5596 in / 1030 out tokens · 23237 ms · 2026-06-25T22:12:33.947484+00:00 · methodology

0 comments

read the original abstract

The notion of $L_\infty$ spaces over dg manifolds is developed. An equivalence between the category of transitive $L_\infty$ algebroids and that of $L_\infty$ spaces is established, and this equivalence detects weak equivalences. Moreover, a faithful functor from $L_\infty$ algebroids to $L_\infty$ spaces is constructed, which also detects weak equivalences.

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 3 canonical work pages

[1]

Alexandrov, A

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work page arXiv 1997
[2]

Grady and O

[GG14] R. Grady and O. Gwilliam, One-dimensional Chern–Simons theory and the Â genus , Algebraic & Geometric Topology 14 (2014), no. 4, 2299–2377. 45 [GG15] , L∞ spaces and derived loop spaces, New York Journal of Mathematics 21 (2015), 231–272. [GG20] , Lie algebroids as spaces , Journal of the Institute of Mathematics of Jussieu 19 (2020), no. 2, 487–53...

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Lada and M

[LM95] T. Lada and M. Markl, Strongly homotopy Lie algebras , Communications in algebra 23 (1995), no. 6, 2147–2161. [LGLS20] C. Laurent-Gengoux, S. Lavau, and T. Strobl, The universal Lie ∞-algebroid of a singular folia- tion, Doc. Math 25 (2020), no. 1571-1652, 1806–00475. [LS19] H.-Y. Liao and M. Stiénon, Formal exponential map for graded manifolds , I...

work page arXiv 1995
[4]

Quillen, Projective modules over polynomial rings , Inventiones Mathematicae 36 (1976), no

[Qui76] D. Quillen, Projective modules over polynomial rings , Inventiones Mathematicae 36 (1976), no. 1, 167–171. [SSS12] H. Sati, U. Schreiber, and J. Stasheff, Twisted differential string and fivebrane structures , Com- munications in Mathematical Physics 315 (2012), no. 1, 169–213. [SSX22] S. Seol, M. Stiénon, and P. Xu, Dg manifolds, formal exponenti...

work page arXiv 1976

[1] [1]

Alexandrov, A

[ASZK97] M. Alexandrov, A. Schwarz, O. Zaboronsky, and M. Kontsevich, The geometry of the master equation and topological quantum field theory , International Journal of Modern Physics A 12 (1997), no. 07, 1405–1429. [BCSX20] R. Bandiera, Z. Chen, M. Stiénon, and P. Xu, Shifted derived Poisson manifolds associated with Lie pairs , Communications in Mathem...

work page arXiv 1997

[2] [2]

Grady and O

[GG14] R. Grady and O. Gwilliam, One-dimensional Chern–Simons theory and the Â genus , Algebraic & Geometric Topology 14 (2014), no. 4, 2299–2377. 45 [GG15] , L∞ spaces and derived loop spaces, New York Journal of Mathematics 21 (2015), 231–272. [GG20] , Lie algebroids as spaces , Journal of the Institute of Mathematics of Jussieu 19 (2020), no. 2, 487–53...

2014

[3] [3]

Lada and M

[LM95] T. Lada and M. Markl, Strongly homotopy Lie algebras , Communications in algebra 23 (1995), no. 6, 2147–2161. [LGLS20] C. Laurent-Gengoux, S. Lavau, and T. Strobl, The universal Lie ∞-algebroid of a singular folia- tion, Doc. Math 25 (2020), no. 1571-1652, 1806–00475. [LS19] H.-Y. Liao and M. Stiénon, Formal exponential map for graded manifolds , I...

work page arXiv 1995

[4] [4]

Quillen, Projective modules over polynomial rings , Inventiones Mathematicae 36 (1976), no

[Qui76] D. Quillen, Projective modules over polynomial rings , Inventiones Mathematicae 36 (1976), no. 1, 167–171. [SSS12] H. Sati, U. Schreiber, and J. Stasheff, Twisted differential string and fivebrane structures , Com- munications in Mathematical Physics 315 (2012), no. 1, 169–213. [SSX22] S. Seol, M. Stiénon, and P. Xu, Dg manifolds, formal exponenti...

work page arXiv 1976