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 CGS
- (A) for an arbitrary prestack A, we have to introduce a differential with an infinite sequence of components instead of just two as in the presheaf case. If A˜ denotes the Grothendieck construction of A, which is a U-graded category, we explicitly construct inverse quasi-isomorphisms F and G between CGS
- (A) and the Hochschild complex CU(A˜), as well as a concrete homotopy T:FG⟶1, 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 L∞-structure on CGS
- (A), which controlls the higher deformation theory of the prestack A. This answers the open problem about the higher structure on the Gerstenhaber-Schack complex at once in the general prestack case.