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 $L_{ \infty} \rightarrow \square p_{ \mathrm{ td}}$. 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.
Categories: math.AT math.KT