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Article

The Gerstenhaber–Schack complex for prestacks

Advances in Mathematics. 2018. Vol. 330. P. 173-228.
Van H. D., Lowen W.

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.