Box operads and higher Gerstenhaber brackets

Abstract

We introduce a symmetric operad $\square p$ ("box-op") which describes a certain calculus of rectangular labeled ``boxes''. Algebras over $\square p$, which we call box operads, have appeared under the name of fc multicategories in work by Leinster \cite{LeinsterFcmulticategories1999}. In our main result, we endow a suitable (graded, zero differential) totalisation $\square p_{\mathrm{td}}$ with a morphism $L_{\infty }\to \square p_{\mathrm{td}}$. We show that $\square p$ acts on an ${\mathbb{N}}^3$-graded enlargement of the ${\mathbb{N}}^2$-graded Gerstenhaber-Schack object ${\mathbf{C}}_{GS}(\mathbb{A})$ of a quiver $\mathbb{A}$ on a small category from \cite{DinhVanLowen2018}. This action restricts to an $L_{\infty }$-structure on ${\mathbf{C}}_{GS}(\mathbb{A})$ (with zero differential). For an element $\alpha =(m,f,c)\in {\mathbf{C}}_{GS}^2(\mathbb{A})$, the Maurer-Cartan equation holds precisely when $(\mathbb{A},m,f,c)$ is a lax prestack with multiplications $m$, restrictions $f$, and twists $c$. As a consequence, the $\alpha$-twisted $L_{\infty }$-structure on ${\mathbf{C}}_{GS}(\mathbb{A})$ controls the deformation theory of $(\mathbb{A},\alpha )$ as a lax prestack.

22 pages + 8 appendix. Minor changes, references updated