Operadic structure on the Gerstenhaber-Schack complex for prestacks

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 $\mathrm{Quilt}$ 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 $\mathrm{Quilt}$ 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.

arXiv comment38 pages, To appear in Selecta Math. Corrected typos, added 45 explanatory drawings, revised notations and provided more explicit definitions and explanations, referred section 2.4 to the appendix and replaced it with a generators and relations description, corrected lemma 2.24, the following definitions, constructions and examples are new: 2.5, 2.6, 2.19, 2.22, 2.29, 3.1, 3.19, 3.25, 4.7

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Cited by in corpus (2)

• Operations on the Hochschild Bicomplex of a Diagram of Algebras
• Deformations of categories of coherent sheaves via quivers with relations