process

Deformations of gerbes on smooth manifolds 357• 2-morphisms b W ! , where ; W m 1 ! m 2 are two 1-morphisms, areelements b 2 1 C A ˝k m R A ˝k R such that m 2 ..a/; b/ D m 2 .b; .a//for all a 2 A ˝k R.It follows easily from the above definition and the nilpotency of m R that Def.A/.R/is a 2-groupoid.Note that Def.A/.R/ is non-empty: it contains the trivial deformation, i.e. the starproduct, still denoted m, which is the R-bilinear extension of the product on A.It is clear that the assignment R 7! Def.A/.R/ extends to a functor on the categoryof commutative Artin k-algebras.3.2.3 Star products and the Deligne 2-groupoid. We continue in notations introducedabove. In particular, we are considering an associative unital k-algebra A. Theproduct m 2 C 2 .A/ determines a cochain, still denoted m 2 g 1 .A/ ˝k R, hence theHochschild differential ı D Œm; in g.A/˝k R for any commutativeArtin k-algebra R.Suppose that m 0 is an R-star product on A. Since .m 0 / WD m 0 m D 0 mod m Rwe have .m 0 / 2 g 1 .A/ ˝k m R . Moreover, the associativity of m 0 implies that .m 0 /satisfies the Maurer–Cartan equation, i.e. .m 0 / 2 MC 2 .g.A/ ˝k m R / 0 .It is easy to see that the assignment m 0 7! .m 0 / extends to a functorDef.A/.R/ ! MC 2 .g.A/ ˝k m R /: (3.4)The following proposition is well known (cf. [9], [11], [10]).Proposition 3.2. The functor (3.4) is an isomorphism of 2-groupoids.3.2.4 Star products on sheaves of algebras. The above considerations generalize tosheaves of algebras in a straightforward way.Suppose that A is a sheaf of k-algebras on a space X. Let mW A˝k A ! A denotethe product.An R-star product on A is a structure of a sheaf of an associative algebras on A˝k Rwhich reduces to modulo the maximal ideal m R . The 2-category (groupoid) of Rstar products on A, denoted Def.A/.R/ is defined just as in the case of algebras; weleave the details to the reader.The sheaf of Hochschild cochains of degree n is defined byC n .A/ WD Hom.A˝n ; A/:We have the sheaf of DGLA g.A/ WD C .A/Œ1, and hence the nilpotent DGLA.XI g.A/ ˝k m R / for every commutative Artin k-algebra R concentrated in degrees 1. Therefore, the 2-groupoid MC 2 ..XI g.A/ ˝k m R / is defined.The canonical functor Def.A/.R/ ! MC 2 ..XI g.A/ ˝k m R / defined just as inthe case of algebras is an isomorphism of 2-groupoids.