Box operads and higher Gerstenhaber brackets

TL;DR

This paper introduces a new symmetric operad called box operad, which models a calculus of labeled boxes and acts on a graded Gerstenhaber-Schack object, revealing a link to deformation theory of lax prestacks.

Contribution

It defines the box operad and demonstrates its action on a graded Gerstenhaber-Schack object, establishing an $L_{ extinfty}$-structure that governs deformations of lax prestacks.

Findings

Introduces the symmetric operad $oxp$ ('box-op') for labeled boxes.

Shows $oxp$ acts on a graded enlargement of the Gerstenhaber-Schack object.

Establishes an $L_{ extinfty}$-structure controlling lax prestack deformations.

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 $L_{\infty} \rightarrow \square p_{\mathrm{td}}$. 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.