Operadic structure on the Gerstenhaber-Schack complex for prestacks

TL;DR

This paper develops an operadic framework for the Gerstenhaber-Schack complex of prestacks, enabling an $L_{ olinebreak}_{ ext{infinity}}$-structure that governs their deformation theory, extending previous work on presheaves.

Contribution

It introduces a new operad acting on the Gerstenhaber-Schack complex of prestacks, extending the $ ext{Quilt}$ operad approach to accommodate twists and prestacks.

Findings

Constructed an operad that acts on the Gerstenhaber-Schack complex for prestacks.

Established an $L_{ ext{infinity}}$-structure governing prestack deformations.

Extended the $ ext{Quilt}$ operad method to the prestack setting with twists.

Abstract

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