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arxiv: 1906.11977 · v1 · pith:SK7NRKOYnew · submitted 2019-06-27 · 🧮 math.AT · math.CT· math.QA

Comparison of spaces associated to DGLA via higher holonomy

Paul Bressler , Alexander Gorokhovsky , Ryszard Nest , Boris Tsygan This is my paper

Pith reviewed 2026-05-25 13:30 UTC · model grok-4.3

classification 🧮 math.AT math.CTmath.QA
keywords differential graded Lie algebraDeligne 2-groupoidHinich simplicial setnon-abelian multiplicative integrationnerve equivalencehigher holonomy
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The pith

For nilpotent DGLAs vanishing below degree -1, the Deligne 2-groupoid nerve is equivalent to Hinich's simplicial set of forms.

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

The paper constructs an explicit equivalence between two models for the space associated to a nilpotent differential graded Lie algebra. One model is the nerve of the Deligne 2-groupoid. The other is the simplicial set of differential forms with values in the Lie algebra, as introduced by Hinich. The maps between them are built using non-abelian multiplicative integration. A reader would care because the result lets the two models be used interchangeably when working with higher structures attached to such algebras.

Core claim

For a nilpotent differential graded Lie algebra whose components vanish in degrees below -1 we construct an explicit equivalence between the nerve of the Deligne 2-groupoid and the simplicial set of differential forms with values in the Lie algebra introduced by V. Hinich. The construction uses the theory of non-abelian multiplicative integration.

What carries the argument

Non-abelian multiplicative integration, which supplies the explicit maps realizing the equivalence between the two simplicial sets.

If this is right

  • Results proved in one model transfer directly to the other.
  • Homotopy invariants computed via forms equal those computed via the groupoid nerve.
  • The equivalence supplies a concrete comparison tool between groupoid and forms constructions for these algebras.

Where Pith is reading between the lines

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

  • The construction might extend to non-nilpotent cases if a suitable completion or localization is introduced.
  • The maps could be used to compare this pair of models with other simplicial or groupoid models appearing in rational homotopy theory.
  • Explicit formulas from the integration might yield new computational methods for deformation problems encoded by such DGLAs.

Load-bearing premise

The differential graded Lie algebra must be nilpotent and have vanishing components in all degrees below -1.

What would settle it

A concrete nilpotent DGLA meeting the degree condition whose two associated simplicial sets have different homotopy groups would disprove the claimed equivalence.

read the original abstract

Fof a nilpotent differential graded Lie algebra whose components vanish in degrees below -1 we construct an explicit equivalence between the nerve of the Deligne 2-groupoid and the simplicial set of differential forms with values in the Lie algebra introduced by V.Hinich. The construction uses the theory of non-abelian multiplicative integration.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript constructs an explicit equivalence, for nilpotent differential graded Lie algebras vanishing in degrees below -1, between the nerve of the Deligne 2-groupoid and Hinich's simplicial set of L-valued differential forms. The construction proceeds via non-abelian multiplicative integration and is stated to be simplicial and bijective on simplices under the given nilpotency and degree restrictions.

Significance. If correct, the result supplies a concrete, explicit bridge between two models for the simplicial space associated to a DGLA, one arising from the Deligne 2-groupoid and the other from Hinich's forms. The explicitness of the map via multiplicative integration is a genuine strength that could support direct computations and comparisons in rational homotopy theory and deformation theory.

minor comments (3)
  1. [Introduction / Main Theorem] The main theorem statement (presumably Theorem 1.1 or equivalent) would benefit from an explicit sentence confirming that the constructed map is a simplicial map, i.e., commutes with all face and degeneracy operators; currently this appears only implicitly through the integration procedure.
  2. [Section 3] Notation for the non-abelian multiplicative integral (e.g., the symbol or operator used for the higher holonomy) should be introduced once and used consistently; occasional switches between integral notation and exponential notation obscure the steps.
  3. [Section 2] A short remark on the necessity of the degree-vanishing hypothesis (components below -1) would help readers; while the restriction is stated, its role in ensuring convergence or well-definedness of the integration is not spelled out.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, recognition of the explicit construction via multiplicative integration, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states an explicit construction of an equivalence between two objects (nerve of Deligne 2-groupoid and Hinich's simplicial set of L-valued forms) for nilpotent DGLAs vanishing in degrees below -1, using non-abelian multiplicative integration. The restriction is declared at the outset; no equations, definitions, or self-citations in the provided abstract or claim reduce the equivalence to a fitted input, self-definition, or load-bearing prior result by the same authors. The derivation is presented as a direct construction inside the restricted regime and remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no free parameters, ad-hoc axioms, or invented entities are introduced; the work relies on standard background in differential graded Lie algebras and simplicial sets.

pith-pipeline@v0.9.0 · 5582 in / 1031 out tokens · 27111 ms · 2026-05-25T13:30:44.830886+00:00 · methodology

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Reference graph

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9 extracted references · 9 canonical work pages · 1 internal anchor

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