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360 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganTherefore D . 0 ; 1 ; 2 / 2 Stack.G ˝ m/, and Ç D .Ç 1 ; Ç 2 / defines a 1-morphism ! .3.4 Acyclicity and strictnessDefinition 3.7. A G-stack . 0 ; 1 ; 2 / is called strict if @ 0 0 0 D @ 0 1 0 , 1 D Id and 2 D Id.Let Stack str .G/ 0 denote the subset of strict G-stacks.Lemma 3.8. Stack str .G/ 0 D MC 2 .ker.G 0 G 1 // 0 .Definition 3.9. For elements 1 ; 2 in Stack str .G/ 0 ,a1-morphism Ç D .Ç 1 ; Ç 2 / 2Stack.G/ 1 . 1 ; 2 / is called strict if @ 0 0 .Ç1 / D @ 0 1 .Ç1 / and Ç 2 D Id.For 1 ; 2 2 Stack str .G/ 0 we denote by Stack str .G/ 1 . 1 ; 2 / the subset of strictmorphisms.Lemma 3.10. For 1 ; 2 2 Stack str .G/ 0 one hasStack str .G/ 1 . 1 ; 2 / D MC 2 .ker.G 0 G 1 // 1 :For 1 ; 2 2 Stack str .G/ 0 let Stack str .G/. 1 ; 2 / denote the full subcategory ofStack.G/. 1 ; 2 / with objects Stack str .G/ 1 . 1 ; 2 /.Thus, we have the 2-groupoids Stack.G/ and Stack str .G/ and an embedding of thelatter into the former which is fully faithful on the respective groupoids of 1-morphisms.Lemma 3.11. Stack str .G/ D MC 2 .ker.G 0 G 1 //.Suppose that G is a cosimplicial DGLA. For each n and i we have the vectorspace G n;i , namely the degree i component of G n . The assignment n 7! G n;i is acosimplicial vector space G ;i .We will consider the following acyclicity condition on the cosimplicial DGLA G:H p .G ;i / D 0 for all i 2 Z and for p ¤ 0. (3.8)Theorem 3.12. Suppose that G is a cosimplicial DGLA which satisfies the condition(3.8), and m a commutative nilpotent ring. Then, the functor W Stack str .G ˝ m/ !Stack.G ˝ m// is an equivalence.Proof. As we already noted before, it is immediate from the definitions that if 1 , 2 2 Stack str .G ˝ m/ 0 , and Ç 1 ; Ç 2 2 Stack str .G ˝ m/ 1 . 1 ; 2 /, then the mappingW Stack str .G ˝ m/ 2 .Ç 1 ; Ç 2 / ! Stack str .G ˝ m/ 2 .Ç 1 ; Ç 2 / is a bijection.Let now 1 ; 2 2 Stack str .G ˝ m/ 0 , Ç D .Ç 1 ; Ç 2 / 2 Stack.G ˝ m/ 1 . 1 ; 2 /.Weshow that there exists È 2 Stack str .G/ 1 . 1 ; 2 / and a 2-morphism W Ç ! È. LetÇ 2 D exp @ 00 20 g, where g 2 .G 1 ˝ m/. Then @ 1 0 g @1 1 g C @1 2 g D 0 mod m2 .As a consequence of the acyclicity condition there exists a 2 .G 0 ˝ m/ such that@ 0 0 a @0 1 a D g mod m2 . Set 1 D exp 0 a. Define then È 1 D . 1 Ç 1 ;.@ 02 1 1 ˝ Id/ ı