A Deligne conjecture for prestacks

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.

To appear in Proceedings of the AMS