arxiv: 2605.13792 · v1 · submitted 2026-05-13 · 🧮 math.AG · math.SG

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A Derived Legendrian Category for Shifted Contact Stacks

Efe \.Izbudak, Kadri \.Ilker Berktav

Authors on Pith no claims yet

Pith reviewed 2026-05-14 17:42 UTC · model grok-4.3

classification 🧮 math.AG math.SG

keywords derived Legendrian categoryn-shifted contact stacksLegendrian correspondencesAKSZ constructionequivariant descentLegendrian surgery(∞,2)-categoriesderived algebraic geometry

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The pith

The derived Legendrian category F_c(X) is constructed for any n-shifted contact derived Artin stack X using Legendrian correspondences.

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

The paper constructs the derived Legendrian category F_c(X) for an n-shifted contact derived Artin stack X. Objects in this category are Legendrian morphisms, while morphism spaces and their composition are defined using equivariant descent. The construction embeds F_c(X) into an (∞,2)-category of spans coming from the AKSZ construction. This also allows topological cobordisms to be viewed as Lagrangian correspondences, thereby defining derived Legendrian surgery. Such a category offers a framework for studying contact structures and their Legendrian aspects within derived algebraic geometry and moduli problems.

Core claim

We construct the derived Legendrian category F_c(X) for an n-shifted contact derived Artin stack X and the (∞,2)-category Leg_n of Legendrian correspondences. The objects are Legendrian morphisms, and the morphism spaces and composition are defined using equivariant descent. The category embeds into an (∞,2)-category of spans defined by the AKSZ construction, and topological cobordisms are evaluated as Lagrangian correspondences to define derived Legendrian surgery.

What carries the argument

The derived Legendrian category F_c(X), whose objects are Legendrian morphisms to X, with morphism spaces and composition supplied by equivariant descent and whose structure embeds into the (∞,2)-category of AKSZ spans.

Load-bearing premise

n-shifted contact structures exist on derived Artin stacks and equivariant descent is well-defined for the relevant morphism spaces and compositions.

What would settle it

A specific n-shifted contact derived Artin stack X where equivariant descent fails to produce associative composition in F_c(X) or where the embedding into the AKSZ span category does not preserve the Legendrian structure.

read the original abstract

We construct the derived Legendrian category $\mathcal{F}_{c}(X)$ for an $n$-shifted contact derived Artin stack $X$ and the $(\infty,2)$-category $Leg_n$ of Legendrian correspondences in the context of derived algebraic geometry, with several applications to moduli theory. In brief, the objects of the category $\mathcal{F}_{c}(X)$ are Legendrian morphisms; the morphism spaces and composition operations are defined using equivariant descent. We also establish that $\mathcal{F}_{c}(X)$ embeds into an $(\infty, 2)$-category of spans defined by the AKSZ construction. We further evaluate topological cobordisms as Lagrangian correspondences to define derived Legendrian surgery.

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 builds a derived Legendrian category F_c(X) for n-shifted contact stacks via equivariant descent and embeds it into AKSZ spans, with surgery defined through cobordisms, but the work stays at the level of a formal construction without visible examples or checks.

read the letter

The core new piece is the definition of F_c(X) whose objects are Legendrian morphisms and whose mapping spaces and composition come from equivariant descent on the shifted contact stack X. They also get an embedding of this category into the (∞,2)-category of spans coming from the AKSZ construction and turn topological cobordisms into Lagrangian correspondences to set up a derived version of Legendrian surgery. That package is the actual output; nothing earlier in the literature they cite does exactly this combination for contact stacks in derived algebraic geometry. The construction looks like a straightforward extension of standard descent and AKSZ techniques, which is a point in its favor if the details hold up. The abstract presents the steps cleanly and avoids obvious circularity. On the other hand, the text gives no concrete examples, no low-dimensional test cases, and no explicit verification that the descent really produces well-behaved morphism spaces in the derived setting. The existence of n-shifted contact structures on the Artin stacks is taken as given, which narrows the range of immediate applications. Without seeing how the proofs handle the higher categorical coherence or whether the embedding is fully faithful, it is difficult to judge how much new mileage this actually gives for moduli problems. The paper is aimed at people already working in derived symplectic or contact geometry who want a categorical language for Legendrian data. It is specialized enough that most readers outside that circle will not need it soon. Still, the construction is coherent on its own terms and the authors engage honestly with the relevant (∞,2)-categorical machinery, so it is worth sending out for refereeing. A serious editor should let it go to review rather than desk-reject; the referees can check the descent details and ask for examples.

Referee Report

0 major / 3 minor

Summary. The manuscript constructs the derived Legendrian category F_c(X) for an n-shifted contact derived Artin stack X, with objects given by Legendrian morphisms and morphism spaces/composition defined via equivariant descent. It also defines the (∞,2)-category Leg_n of Legendrian correspondences, proves that F_c(X) embeds into an (∞,2)-category of spans arising from the AKSZ construction, and interprets topological cobordisms as Lagrangian correspondences to define derived Legendrian surgery, with applications to moduli theory.

Significance. If the constructions hold, the work supplies a higher-categorical framework linking shifted contact structures on derived Artin stacks with Legendrian correspondences and AKSZ spans. This could furnish new tools for moduli problems in derived contact geometry and for surgery-type operations in the derived setting, extending standard descent and AKSZ techniques without introducing free parameters or ad-hoc axioms.

minor comments (3)

[§1] §1: The precise definition of 'Legendrian morphism' into an n-shifted contact stack X should be stated explicitly before the descent construction is invoked, to make the objects of F_c(X) unambiguous.
[§3.2] §3.2: The embedding of F_c(X) into the AKSZ span category is stated as a fully faithful functor; a brief diagram or explicit description of the image would clarify the claim.
The paper would benefit from a short comparison table or paragraph contrasting the new derived Legendrian category with existing (∞,1)-categorical versions of Legendrian contact homology.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. The report provides no specific major comments to address point by point.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper constructs the derived Legendrian category F_c(X) directly from the data of an n-shifted contact derived Artin stack X by taking Legendrian morphisms as objects and defining morphism spaces and composition via equivariant descent; it then embeds the result into an (∞,2)-category of AKSZ spans and interprets topological cobordisms as Lagrangian correspondences for surgery. None of these steps reduces by definition or by self-citation to a fitted parameter, a renamed input, or an unverified uniqueness claim internal to the paper; the construction is presented as an extension of standard descent and AKSZ techniques in derived algebraic geometry and remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the standard framework of derived algebraic geometry and infinity-category theory plus domain assumptions about shifted contact structures; no free parameters or new invented entities with independent evidence are visible from the abstract.

axioms (2)

standard math Standard axioms and constructions of derived algebraic geometry and (∞,2)-categories
Invoked for the definition of stacks, morphisms, descent, and AKSZ spans.
domain assumption Existence of n-shifted contact structures on derived Artin stacks
Required for the objects X on which F_c(X) is defined.

invented entities (1)

Derived Legendrian category F_c(X) no independent evidence
purpose: Category whose objects are Legendrian morphisms on shifted contact stacks
Newly constructed via equivariant descent in the paper.

pith-pipeline@v0.9.0 · 5422 in / 1372 out tokens · 39121 ms · 2026-05-14T17:42:05.182592+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/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
We construct the derived Legendrian category F_c(X) for an n-shifted contact derived Artin stack X ... morphism spaces and composition operations are defined using equivariant descent ... embeds into an (∞,2)-category of spans defined by the AKSZ construction.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
The homotopy pullback Z ≃ L1 ×h_X L2 has an (n-1)-shifted symplectic structure ... descent ... weight 1 G_m-action

Reference graph

Works this paper leans on

8 extracted references · 3 canonical work pages · 2 internal anchors

[1]

Equivariant Quotients of Derived Symplectic Spaces and Legendrian Intersection Theorem

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work page internal anchor Pith review Pith/arXiv arXiv 2026
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T. Pantev, B. To¨en, M. Vaqui´e, G. Vezzosi,Shifted Symplectic Structures, Publ. Math. Inst. Hautes ´Etudes Sci. 117 (2013), 271-328

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Calaque, R

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work page arXiv 2022
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K. ˙I. Berktav,Shifted Contact Structures and Their Local Theory, Ann. Fac. Sci. Toulouse, Math., Serie 6, Vol. 33(4): 1019-1057, 2024

2024
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K. ˙I. Berktav,On shifted contact derived Artin stacks, Higher Structures 9(2):103–135, 2025

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work page internal anchor Pith review Pith/arXiv arXiv 2000