A Deligne conjecture for prestacks

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