标题： 一个预堆栈的德利涅猜想 显示英文标题

标题： A Deligne conjecture for prestacks

摘要： 我们证明了预堆栈的Deligne猜想的类似结论。 我们表明，给定一个预堆栈$\mathbb A$，其Gerstenhaber--Schack复形$\mathbf{C}_{\mathsf{GS}}(\mathbb A)$自然是一个$E_2$-代数。 这种结构推广了已知的$\mathsf{L}_\infty$-代数结构在$\mathbf{C}_{\mathsf{GS}}(\mathbb A)$上的结构，以及其上同调$\mathbf{H}_{\mathsf{GS}}(\mathbb A)$上的Gerstenhaber代数结构。 主要成分是证明了Hawkins的猜想\cite{hawkins}，该猜想指出微分分次操作符$\mathsf{Quilt}$在正次数上同调为零。 作为推论，$\mathsf{Quilt}$与编码括号代数的操作符$\mathsf{Brace}$是拟同构的。 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. 显示英文摘要

摘要： 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.