process

Deformations of gerbes on smooth manifolds 373For W Œn ! letG.A/ D .N .n/ UI .0n/ C .Mat.A/ .0/ / loc Œ1/:Suppose given another simplex W Œm ! and morphism W Œm ! Œn suchthat D ı (i.e. is a morphism of simplices ! ). The morphism .0n/ factorsuniquely into 0 ! .0/ ! .m/ ! n, which, under , gives the factorization of.0n/W .0/ ! .n/ (in ) intowhere g D .0m/. The mapis the composition.0/ f ! .0/ g ! .m/ h ! .n/; (5.7) W G.A/ ! G.A/ .N .m/ UI g C .Mat.A/ .0/ / loc Œ1/h ! .N .n/ UI h g C .Mat.A/ .0/ / loc Œ1/f ] ! .N .n/ UI h g f C .Mat.A/ .0/ / loc Œ1/:Suppose given yet another simplex, W Œl ! , and a morphism of simplicesW ! , i.e. a morphism W Œl ! Œm such that D ı . Then, the composition ı W G.A/ ! G.A/ coincides with the map . ı / .For n D 0;1;2;:::letG.A/ n DYG.A/ : (5.8)Œn !A morphism W Œm ! Œn in induces the map of DGLA W G.A/ m ! G.A/ n .The assignment 3 Œn 7! G.A/ n , 7! defines the cosimplicial DGLA denotedby G.A/.5.3 AcyclicityTheorem 5.2. The cosimplicial DGLA G.A/ is acyclic, i.e. it satisfies the condition(3.8).The rest of the section is devoted to the proof of Theorem 5.2. We fix a degree ofHochschild cochains k.For W Œn ! let c D .N .n/ UI .0n/ C k .Mat.A/ .0/ / loc /. For a morphism W ! we have the map W c ! c defined as in 5.2.3.Let .C ;@/denote the corresponding cochain complex whose definition we recallbelow. For n D 0;1;::: let C n D Q Œn !c . The differential @ n W C n ! C nC1 isdefined by the formula @ n D P nC1iD0 . 1/i .@ n i / .