process

Deformations of gerbes on smooth manifolds 359Proof. The proof can be obtained by applying Theorem 3.1 repeatedly.Let 1 , 2 2 Stack.G ˝ m/ 0 , and let Ç 1 , Ç 2 be two 1-morphisms between 1and 2 . Note that W MC 2 .G 0 ˝ m/ 2 .Ç 1 1 ; Ç1 2 / ! MC2 .H 0 ˝ m/ 2 . Ç 1 1 ; Ç 1 2 / isa bijection by Theorem 3.1. Injectivity of the map W Stack.G ˝ m/ 2 .Ç 1 ; Ç 2 / !Stack.H ˝ m/ 2 . Ç 1 ; Ç 2 / follows immediately.For the surjectivity, note that an element of Stack.H˝m/ 2 . Ç 1 ; Ç 2 / is necessarilygiven by for some 2 MC 2 .G 0˝m/.Ç 1 1 ; Ç1 2 / 2 and the following identity is satisfiedin MC 2 .H 1 ˝ m/. . 1 2 ı @0 0 Ç1 1 /; .@ 0 1 Ç1 2 ı 1 1 // 2: .Ç 2 2 ı .@1 0 ˝ Id// D ..Id ˝ @ 1 1 / ı Ç2 1 /:Since W MC 2 .G 1 ˝ m/. 1 2 ı @0 0 Ç1 1 ;@0 1 Ç1 2 ı 1 1 / 2 ! MC 2 .H 1 ˝ m/. . 1 2 ı @0 0 Ç1 1 /; .@ 0 1 Ç1 2 ı 1 1 // 2 is bijective, and in particular injective, Ç 2 2 ı.@1 0 ˝Id/ D .Id˝@1 1 /ıÇ2 1 ,and defines an element in Stack.G ˝ m/ 2 .Ç 1 ; Ç 2 /.Next, let 1 , 2 2 Stack.G ˝ m/ 0 , and let Ç be a 1-morphisms between 1and 2 . We show that there exists È 2 Stack.G ˝ m/ 1 . 1 ; 2 / such that È isisomorphic to Ç. Indeed, by Theorem 3.1, there exists È 1 2 MC 2 .G 0 ˝ m/ 1 . 1 ; 2 /such that MC 2 .H 0˝m/ 2 . È 1 ; Ç/ ¤;. Let 2 MC 2 .H 0˝m/ 2 . È 1 ; Ç/. Define 2MC 2 .H 1 ˝m/ 2 . . 1 2 ı@0 0 È1 /; .@ 0 1 È1 ı 1 1 // by D .@0 1 ˝Id/ 1 ıÇ 2 ı.Id ˝@ 0 0 /.It is easy to verify that the following identities holds:.Id ˝ 1 2 / ı .@1 2 ˝ Id/ ı .Id ˝ @1 0 / D @1 1 ı . 2 2 ˝ Id/;s0 1 D Id:By bijectivity of on MC 2 there exists a unique È 2 2 MC 2 .H 1˝m/ 2 .2 1ı@0 0 È1 ;@ 0 1 È1 ı1 1/ such that È 2 D . Moreover, as before, injectivity of implies that theconditions (3.6) are satisfied. Therefore È D .È 1 ; È 2 / defines a 1-morphism 1 ! 2and is a 2-morphism È ! Ç.Now, let 2 Stack.H ˝ m/ 0 . We construct 2 Stack.G ˝ m/ 0 in such a way thatStack.H˝m/ 1 . ;/ ¤ ¿. By Theorem 3.1 there exists 0 2 MC 2 .G 0 ˝m/ 0 suchthat MC 2 .H 0 ˝ m/ 1 . 0 ; 0 / ¤ ¿. Let Ç 1 2 MC 2 .H 0 ˝ m/ 1 . ;/. ApplyingTheorem 3.1 again we obtain that there exists 2 MC 2 .G 1 ˝ m/ 1 .@ 0 0 0 ;@ 0 1 0 / suchthat there exists 2 MC 2 .H 1 ˝ m/ 2 . 1 ı @ 0 0 Ç1 ;@ 0 1 Ç1 ı /. Then s0 1 is then a2-morphism Ç 1 ! Ç 1 ı .s0 1/, which induces a 2-morphism W .s1 0 / 1 ! Id.As a next step set 1 D ı .@ 0 0 .s1 0 // 1 2 MC 2 .G 1 ˝ m/ 1 .@ 0 0 0 ;@ 0 1 0 /, Ç 2 D˝.@ 0 0 / 1 2 MC 2 .H 1˝m/ 2 . 1 ı@ 0 0 Ç1 ;@ 0 1 Ç1 ı 1 /. It is easy to see that s0 11 D Id,s0 1Ç2 D Id.We then conclude that there exists a unique 2 such that.Id ˝ 2 / ı .@ 1 2 Ç2 ˝ Id/ ı .Id ˝ @ 1 0 Ç2 / D @ 1 1 Ç2 ı . 2 ˝ Id/:Such a 2 necessarily satisfies the conditions@ 2 2 2 ı .Id ˝ @ 2 0 2 / D @ 2 1 2 ı .@ 2 3 2 ˝ Id/;s0 2 2 D s1 2 2 D Id: