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358 P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan3.3 G-stacks. Suppose that G W Œn ! G n is a cosimplicial DGLA. We assume thateach G n is a nilpotent DGLA. We denote its component of degree i by G n;i and assumethat G n;i D 0 for i< 1.Definition 3.3. A G-stack is a triple D . 0 ; 1 ; 2 /, where• 0 2 MC 2 .G 0 / 0 ,• 1 2 MC 2 .G 1 / 1 .@ 0 0 0 ;@ 0 1 0 /,satisfying the conditions 1 0 1 D Id;• 2 2 MC 2 .G 2 / 2 .@ 1 2 . 1 / ı @ 1 0 . 1 /; @ 1 1 . 1 //satisfying the conditions@ 2 2 2 ı .Id ˝ @ 2 0 2/ D @ 2 1 2 ı .@ 2 3 2 ˝ Id/; and s 2 0 2 D s 2 1 2 D Id: (3.5)Let Stack.G/ 0 denote the set of G-stacks.Definition 3.4. For 1 ; 2 2 Stack.G/ 0 a 1-morphism ÇW 1 ! 2 is a pair Ç D.Ç 1 ; Ç 2 /, where Ç 1 2 MC 2 .G 0 / 1 . 0 1 ;0 2 /, Ç2 2 MC 2 .G 1 / 2 . 1 2 ı @0 0 .Ç1 /; @ 0 1 .Ç1 / ı 1 1 /,satisfying.Id ˝ 1 2 / ı .@1 2 Ç2 ˝ Id/ ı .Id ˝ @ 1 0 Ç2 / D @ 1 1 Ç2 ı .2 2 ˝ Id/;s0 1 Ç2 D Id:(3.6)Let Stack.G/ 1 . 1 ; 2 / denote the set of 1-morphisms 1 ! 2 .Composition of 1-morphisms ÇW 1 ! 2 and ÈW 2 ! 3 is given by .È 1 ı Ç 1 ;.Ç 2 ˝ Id/ ı .Id ˝ È 2 //.Definition 3.5. For Ç 1 ; Ç 2 2 Stack.G/ 1 . 1 ; 2 / a 2-morphism W Ç 1 ! Ç 2 is a 2-morphism 2 MC 2 .G 0 / 2 .Ç 1 1 ; Ç1 2/ which satisfiesÇ 2 2 ı .Id ˝ @0 0 / D .@0 1 ˝ Id/ ı Ç2 1 : (3.7)Let Stack.G/ 2 .Ç 1 ; Ç 2 / denote the set of 2-morphisms.Compositions of 2-morphisms are given by the compositions in MC 2 .G 0 / 2 .For 1 ; 2 2 Stack.G/ 0 , we have the groupoid Stack.G/. 1 ; 2 / with the set ofobjects Stack.G/ 1 . 1 ; 2 / and the set of morphisms Stack.G/ 2 .Ç 1 ; Ç 2 / under verticalcomposition.Every morphism of cosimplicial DGLA induces in an obvious manner a functor W Stack.G ˝ m/ ! Stack.H ˝ m/We have the following cosimplicial analogue of Theorem 3.1:Theorem 3.6. Suppose that W G ! H is a quasi-isomorphism of cosimplicial DGLAand m is a commutative nilpotent ring. Then the induced map W Stack.G ˝ m/ !Stack.H ˝ m/ is an equivalence.