process

Deformations of gerbes on smooth manifolds 361Ç 2 ı .Id ˝ @ 0 0 1/ 1 / 2 Stack.G ˝ m/ 1 . 1 ; 2 /. Note that 1 defines a 2-morphismÇ ! È 1 . Note also that È 2 1 2 exp @ 0 0 2 0 .G ˝m 2 /. Proceeding inductively one constructsa sequence È k 2 Stack.G ˝ m/ 1 . 1 ; 2 / such that È 2 k 2 exp.G ˝ mkC1 / and 2-morphisms k W Ç ! È k , kC1 D k mod m k . Since m is nilpotent, for k large enoughwe have È k 2 Stack str .G/ 1 . 1 ; 2 /.Assume now that 2 Stack.G ˝ m/ 0 . We will construct 2 Stack str .G ˝ m/ 0and a 1-morphism ÇW ! .We begin by constructing 2 Stack.G ˝m/ 0 such that 2 D Id and a 1-morphismÈW ! . Wehave: 2 D exp @ 11 @ 0 1 c, c 2 .G 2 ˝ m/. In view of the equations (3.5)0c satisfies the identities@ 2 0 c @2 1 c C @2 2 c @2 3 c D 0 mod m2 ; and s0 2 c D s2 1c D 0: (3.9)By the acyclicity of the normalized complex we can find b 2 G 1 ˝ m such that@ 1 0 b @1 1 b C @1 2 b D c mod m2 , s0 1b D 0. Let 1 D exp @ 01 . b/. Define then0 1 D . 0 1 ;1 1 ;2 1 / where 0 1 D 0 , 1 1 D 1 1 , and 2 1is such that.Id ˝ 2 / ı .@ 1 2 1 ˝ Id/ ı .Id ˝ @ 1 0 1/ D @ 1 1 1 ı . 2 1 ˝ Id/:Note that 2 1 D Id mod m2 and .Id; 1 / is a 1-morphism ! 1 . As beforewe can construct a sequence k such that 2 kD Id mod mkC1 , and 1-morphisms.Id; k /W ! k , kC1 D k mod m k . As before we conclude that this gives thedesired construction of . The rest of the proof, i.e. the construction of is completelyanalogous.Corollary 3.13. Suppose that G is a cosimplicial DGLA which satisfies the condition(3.8). Then there is a canonical equivalence:Stack.G ˝ m/ Š MC 2 .ker.G 0 G 1 / ˝ m/:Proof. Combine Lemma 3.11 with Theorem 3.12.4 Algebroid stacksIn this section we review the notions of algebroid stack and twisted form. We alsodefine the notion of descent datum and relate it with algebroid stacks.4.1 Algebroids and algebroid stacks4.1.1 Algebroids. For a category C we denote by iC the subcategory of isomorphismsin C; equivalently, iC is the maximal subgroupoid in C.Suppose that R is a commutative k-algebra.Definition 4.1. An R-algebroid is a nonempty R-linear category C such that thegroupoid iC is connected