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A Deligne conjecture for prestacks

Ricardo Campos, Lander Hermans

TL;DR

The paper constructs an explicit $E_2$-algebra action on the Gerstenhaber–Schack complex of a prestack by introducing the twisting quilt operad $ ext{TwQuilt}$, which is quasi-isomorphic to both the little disks and homotopy Gerstenhaber operads. It proves Hawkins' conjecture that the Quilt operad has vanishing positive homology, establishing a quasi-isomorphism $ ext{Quilt} o ext{Brace}$, and further elevates the algebraic structure through a $ ext{PreLie}_ ext{∞}$ lift that yields a robust $E_2$-model after twisting. The action of $ ext{TwQuilt}$ on the desuspended GS complex $ extbf{C}_{ ext{GS}}( ext{A})$ provides an explicit mechanism to lift the Gerstenhaber structure to the level of the complex, ensuring a Gerstenhaber algebra on GS cohomology and guiding deformations of prestacks. In the presheaf special case, an additional orthogonal $ ext{TwQuilt}$-action recovers Hawkins’ framework with a corresponding $L_ ext{∞}$-structure, thereby giving a comprehensive, explicit solution to Deligne’s conjecture for prestacks and their deformations.

Abstract

We prove an analog of the Deligne conjecture for prestacks. We show that given a prestack $\mathbb A$, its Gerstenhaber--Schack complex $\mathbf{C}_{\mathsf{GS}}(\mathbb A)$ is naturally an $E_2$-algebra. This structure generalises both the known $\mathsf{L}_\infty$-algebra structure on $\mathbf{C}_{\mathsf{GS}}(\mathbb A)$, as well as the Gerstenhaber algebra structure on its cohomology $\mathbf{H}_{\mathsf{GS}}(\mathbb A)$. The main ingredient is the proof of a conjecture of Hawkins \cite{hawkins}, stating that the dg operad $\mathsf{Quilt}$ has vanishing homology in positive degrees. As a corollary, $\mathsf{Quilt}$ is quasi-isomorphic to the operad $\mathsf{Brace}$ encoding brace algebras. In addition, we improve the $L_\infty$-structure on $\mathsf{Quilt}$ by showing that it originates from a $\mathsf{PreLie}_\infty$-structure lifting the $\mathsf{PreLie}$-structure on $\mathsf{Brace}$ in homology.

A Deligne conjecture for prestacks

TL;DR

The paper constructs an explicit -algebra action on the Gerstenhaber–Schack complex of a prestack by introducing the twisting quilt operad , which is quasi-isomorphic to both the little disks and homotopy Gerstenhaber operads. It proves Hawkins' conjecture that the Quilt operad has vanishing positive homology, establishing a quasi-isomorphism , and further elevates the algebraic structure through a lift that yields a robust -model after twisting. The action of on the desuspended GS complex provides an explicit mechanism to lift the Gerstenhaber structure to the level of the complex, ensuring a Gerstenhaber algebra on GS cohomology and guiding deformations of prestacks. In the presheaf special case, an additional orthogonal -action recovers Hawkins’ framework with a corresponding -structure, thereby giving a comprehensive, explicit solution to Deligne’s conjecture for prestacks and their deformations.

Abstract

We prove an analog of the Deligne conjecture for prestacks. We show that given a prestack , its Gerstenhaber--Schack complex is naturally an -algebra. This structure generalises both the known -algebra structure on , as well as the Gerstenhaber algebra structure on its cohomology . The main ingredient is the proof of a conjecture of Hawkins \cite{hawkins}, stating that the dg operad has vanishing homology in positive degrees. As a corollary, is quasi-isomorphic to the operad encoding brace algebras. In addition, we improve the -structure on by showing that it originates from a -structure lifting the -structure on in homology.

Paper Structure

This paper contains 19 sections, 16 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. The GS complex
  3. Structure of the proof
  4. The operad $\mathop{\mathrm{\mathsf{Quilt}}}\nolimits$ and its homology
  5. The operad $\mathop{\mathrm{\mathsf{Quilt}}}\nolimits$
  6. The operad $\mathop{\mathrm{\mathsf{Brace}}}\nolimits$
  7. The operad $\mathop{\mathrm{\mathsf{F_{2}Surj}}}\nolimits$
  8. The operad $\mathop{\mathrm{\mathsf{Quilt}}}\nolimits$
  9. The homology of $\mathop{\mathrm{\mathsf{Quilt}}}\nolimits$
  10. The double complex $\mathop{\mathrm{\mathsf{Quilt}}}\nolimits(T)_{\bullet,\bullet}$
  11. The homology of $\mathop{\mathrm{\mathsf{Quilt}}}\nolimits(T)$
  12. The morphism $\mathop{\mathrm{\mathop{\mathrm{\mathsf{PreLie}}}\nolimits_{\infty}}}\nolimits \longrightarrow \mathop{\mathrm{\mathsf{Quilt}}}\nolimits$
  13. The operad $\mathop{\mathrm{\mathop{\mathrm{\mathsf{PreLie}}}\nolimits_{\infty}}}\nolimits$
  14. The morphism $\mathop{\mathrm{\mathop{\mathrm{\mathsf{PreLie}}}\nolimits_{\infty}}}\nolimits \longrightarrow \mathop{\mathrm{\mathsf{Quilt}}}\nolimits$
  15. A new model for $\mathsf{E}_2$
  16. ...and 4 more sections

Key Result

Theorem 1.2

Given a prestack $\mathbb A$, there is an action of $\mathsf{TwQuilt}$ on its Gerstenhaber--Schack complex $\mathbf{C}_{\mathsf{GS}}(\mathbb A)$ inducing a Gerstenhaber algebra structure on Gerstenhaber--Schack cohomology $\mathbf{H}_{\mathsf{GS}}(\mathbb A)$.

Theorems & Definitions (33)

  • Theorem 1.2: Theorem \ref{['thmaction']}
  • Theorem 1.3: Theorem \ref{['thmhomologyQuilt']}
  • Proposition 1.4: Proposition \ref{['propprelieinf']}
  • Theorem 1.5: Corollary \ref{['cormodel']}
  • Theorem 1.6: Theorem \ref{['thmaction']}
  • Example 2.1
  • Lemma 2.2
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • ...and 23 more