The Gerstenhaber–Schack complex for prestacks

Abstract

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.