Résumé
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 \operatorname {\mathsf{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 [Adv. Math. 428 (2023), p. 80], stating that the dg operad \operatorname {\mathsf{Quilt}} has vanishing homology in positive degrees. As a corollary, \operatorname {\mathsf{Quilt}} is quasi-isomorphic to the operad \operatorname {\mathsf{Brace}} encoding brace algebras. In addition, we improve the \operatorname {\mathsf{L}_{\infty }}-structure on \operatorname {\mathsf{Quilt}} by showing that it originates from a \operatorname {\operatorname {\mathsf{PreLie}}_{\infty }}-structure lifting the \operatorname {\mathsf{PreLie}}-structure on \operatorname {\mathsf{Brace}} in homology.