Abstract
We introduce an operad {{,mathrm{Patch},}} 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.
Highlights
The deformation theory of algebras due to Gerstenhaber furnishes the guiding example for algebraic deformation theory
For an algebra A, the Hochschild complex C(A) is a dg Lie algebra governing the deformation theory of A through the Maurer–Cartan formalism. This dg Lie structure is the shadow of a richer operadic structure, which can be expressed by saying that C(A) is a homotopy G-algebra [6]
In [2], Dinh Van and Lowen established a Gerstenhaber–Schack complex for prestacks, involving a differential which features an infinite sequence of higher components in addition to the classical simplicial and Hochshild differentials
Summary
The deformation theory of algebras due to Gerstenhaber furnishes the guiding example for algebraic deformation theory. The deformation theory of algebras was later extended to presheaves of algebras by Gerstenhaber and Schack, who in particular introduced a bicomplex computing the natural bimodule Ext groups [4,5]. This GS-complex C(A) of a presheaf A does not control deformations of A as a presheaf, but rather as a twisted presheaf, see for instance [2,11]. In the case of a presheaf (A, m, f ), in [8], Hawkins introduces an operad Quilt ⊆ F2S ⊗H Brace which he later extends to an operad mQuilt acting on the GS-complex These operads are naturally endowed with L∞-operations as desired. The current paper is part of a larger project in which it is our goal to understand the homotopy equivalence CGS(A) ∼= CC(A!) from [2] operadically, showing in particular that the L∞-structure from [2] and the one established in the present paper coincide
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