The Gerstenhaber–Schack complex for prestacks
The aim of this work is to construct a complex which through its higher structure directly controlls deformations of general prestacks, building on the work of Gerstenhaber and Schack for presheaves of algebras. In defining a Gerstenhaber–Schack complex for an arbitrary prestack , we have to introduce a differential with an infinite sequence of components instead of just two as in the presheaf case. If denotes the Grothendieck construction of , which is a -graded category, we explicitly construct inverse quasi-isomorphisms and between and the Hochschild complex , as well as a concrete homotopy , which had not been obtained even in the presheaf case. As a consequence, by applying the Homotopy Transfer Theorem, one can transfer the dg Lie structure present on the Hochschild complex in order to obtain an -structure on , which controlls the higher deformation theory of the prestack . This answers the open problem about the higher structure on the Gerstenhaber–Schack complex at once in the general prestack case.