Operadic structure on the Gerstenhaber–Schack complex for prestacks

We introduce an operad which acts on the Gerstenhaber-Schack complex of a prestack as defined by Dinh Van and Lowen, in particular allows us to endow this with underlying $L_{\infty}$-structure. make use $\operatorname{Quilt}$ was used Hawkins order solve presheaf case. Due additional difficulty posed presence twists, we have fundamentally different way (even for presheaves) allow extension prestacks. The resulting $L_{\infty}$-algebra governs deformation theory prestack.