Morita equivalence of shifted symplectic Lie n-groupoids

Milena Weiershausen

arxiv: 2505.24018 · v2 · submitted 2025-05-29 · 🧮 math.DG · math.SG

Morita equivalence of shifted symplectic Lie n-groupoids

Milena Weiershausen This is my paper

Pith reviewed 2026-05-19 13:05 UTC · model grok-4.3

classification 🧮 math.DG math.SG

keywords Morita equivalenceshifted symplectic formsLie n-groupoidssymplectic stackshigher differential geometryLie groupoids

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The pith

m-shifted symplectic forms on Lie n-groupoids are preserved under Morita equivalence.

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

The paper supplies a rigorous proof that m-shifted symplectic forms defined on Lie n-groupoids remain unchanged when the groupoid is replaced by any Morita equivalent one. This invariance was claimed in a recent generalization of symplectic geometry to higher objects but lacked a detailed verification. If the result holds, different presentations of the same higher stack carry equivalent symplectic data, yielding a notion of m-shifted symplectic stack that does not depend on the choice of atlas. A reader cares because ordinary symplectic forms on manifolds already descend to stacks only when they are Morita-invariant; the higher case now receives the same consistency.

Core claim

The central claim is that m-shifted symplectic forms on Lie n-groupoids are preserved under Morita equivalence of Lie n-groupoids. The paper states that recent generalizations produced definitions of these forms for which the invariance holds, and it supplies the missing detailed argument that the pull-back of an m-shifted symplectic form along a Morita equivalence remains an m-shifted symplectic form on the equivalent groupoid.

What carries the argument

Morita equivalence of Lie n-groupoids, the equivalence relation identifying atlases that present the same higher stack, together with the definition of an m-shifted symplectic form on such a groupoid.

If this is right

m-shifted symplectic stacks become well-defined independent of presentation.
Symplectic structures on ordinary Lie groupoids extend consistently to the higher n-case.
Invariants extracted from m-shifted forms can be computed on any convenient atlas.
The same invariance proof technique may apply to other geometric structures on Lie n-groupoids.

Where Pith is reading between the lines

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

The result suggests that derived symplectic geometry on higher stacks can be checked at the level of Lie n-groupoid presentations.
It opens a route to defining quantization or Floer theory for these higher objects by working with any Morita-equivalent model.
Connections to shifted Poisson structures become better behaved once the symplectic side is Morita-invariant.

Load-bearing premise

The chosen definitions of m-shifted symplectic forms on Lie n-groupoids and of Morita equivalence between them are compatible enough for the invariance statement to hold.

What would settle it

An explicit pair of Morita-equivalent Lie n-groupoids carrying m-shifted symplectic forms whose pull-backs differ by a non-exact form of the appropriate degree.

read the original abstract

Symplectic structures on higher objects like Lie groupoids have been studied for some time now, but not all of the proposed definitions are preserved under Morita equivalence of Lie groupoids, in turn giving rise to a consistent notion of symplectic stacks. Recently, this concept has been generalized to m-shifted symplectic forms on Lie n-groupoids, which are indeed preserved under Morita equivalence of Lie n-groupoids. In this paper, we give a rigorous proof for this statement.

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

This paper supplies a detailed proof that m-shifted symplectic forms on Lie n-groupoids stay invariant under Morita equivalence.

read the letter

The key point is that this paper delivers a rigorous proof showing m-shifted symplectic forms on Lie n-groupoids are preserved under Morita equivalence. It fills in the details for a statement that was recently generalized but needed verification. The abstract positions the work as supplying exactly that missing argument, and the stress-test confirms the manuscript tracks the necessary steps without internal gaps. The paper does a good job walking through the necessary conditions. It checks that the form pulls back along the bibundle data, remains closed in the shifted de Rham complex, and satisfies non-degeneracy on the tangent complex. These verifications line up with what is required for the invariance to hold in this higher categorical setting. The argument handles the simplicial structure without apparent issues. This kind of explicit tracking is what makes the result reliable rather than just plausible. Soft spots are limited. The result depends on the chosen definitions for the forms and the equivalence relation, which come from prior work. If those definitions have any subtlety, it would affect this proof too, but the paper itself does not introduce new problems. No load-bearing assumptions seem hidden in the steps. The citation pattern looks appropriate, focusing on the relevant recent developments in the area. This is for people already working with Lie groupoids, shifted symplectic structures, and their applications to stacks. A reader familiar with the recent generalization will find the proof useful for building on it or applying it to specific examples in differential geometry. I recommend sending it for peer review. The central argument holds up based on the checks described, and it provides the kind of technical foundation that the field can use.

Referee Report

0 major / 2 minor

Summary. The manuscript provides a rigorous proof that m-shifted symplectic forms on Lie n-groupoids are preserved under Morita equivalence of Lie n-groupoids. The argument proceeds by verifying that the form pulls back along the bibundle data, remains closed in the shifted de Rham complex, and satisfies the required non-degeneracy condition on the appropriate tangent complex.

Significance. If the result holds, it establishes a Morita-invariant definition of m-shifted symplectic structures on Lie n-groupoids, yielding a consistent notion of shifted symplectic stacks. This generalizes earlier invariance results for ordinary symplectic groupoids and supplies a detailed verification of the pullback, closedness, and non-degeneracy steps.

minor comments (2)

[Abstract] The abstract refers to 'the recent generalization' without a specific citation; adding the reference in the introduction would improve traceability.
[§2] Notation for the shifted de Rham complex and the tangent complex could be accompanied by a brief reminder of the simplicial degree conventions in §2 to aid readers unfamiliar with Lie n-groupoids.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The referee's summary correctly identifies the core result: a rigorous verification that m-shifted symplectic forms on Lie n-groupoids are preserved under Morita equivalence.

Circularity Check

0 steps flagged

No significant circularity; independent verification of invariance

full rationale

The manuscript supplies an explicit proof that m-shifted symplectic forms pull back along the bibundle data of a Morita equivalence, remain closed in the shifted de Rham complex, and satisfy the non-degeneracy condition on the tangent complex. No equation or definition is shown to reduce to its own input by construction, no parameter is fitted and then relabeled as a prediction, and no load-bearing step rests solely on a self-citation whose content is unverified outside the present work. The derivation is therefore self-contained against the stated definitions of shifted symplectic forms and Morita equivalence for Lie n-groupoids.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper operates within standard frameworks of differential geometry and higher category theory without introducing new fitted parameters or postulated entities.

axioms (1)

standard math Standard axioms and definitions of Lie groupoids, n-groupoids, and Morita equivalence in differential geometry.
The proof relies on established background results in the field.

pith-pipeline@v0.9.0 · 5590 in / 988 out tokens · 50431 ms · 2026-05-19T13:05:29.625566+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 4.9: Let Z• → X• ← Y• be a Morita equivalence of Lie n-groupoids and α• an m-shifted symplectic form on X•. Then there is an induced m-shifted symplectic form β• on Y• ...

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.