A Deligne conjecture for prestacks

TL;DR

This paper proves an analog of the Deligne conjecture for prestacks, showing their Gerstenhaber--Schack complex has an $E_2$-algebra structure, extending known algebraic structures and confirming a conjecture about the dg operad Quilt.

Contribution

It establishes an $E_2$-algebra structure on the Gerstenhaber--Schack complex of prestacks and proves the vanishing homology of the Quilt operad, confirming a conjecture of Hawkins.

Findings

Gerstenhaber--Schack complex is an $E_2$-algebra

The dg operad Quilt has vanishing homology in positive degrees

Quilt is quasi-isomorphic to the Brace operad

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.