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Deformations of gerbes on smooth manifolds 3512 Preliminaries2.1 Simplicial notions2.1.1 The category of simplices. For n D 0;1;2;:::we denote by Œn the categorywith objects 0;:::;nfreely generated by the graph0 ! 1 !!n:For 0 p q n we denote by .pq/ the unique morphism p ! q.We denote by the full subcategory of Cat with objects the categories Œn forn D 0;1;2;:::.For 0 i n C 1 we denote by @ i D @ n i W Œn ! Œn C 1 the i th face map, i.e. theunique map whose image does not contain the object i 2 Œn C 1.For 0 i n 1 we denote by s i D s n i W Œn ! Œn 1 the i th degeneracy map,i.e. the unique surjective map such that s i .i/ D s i .i C 1/.2.1.2 Simplicial and cosimplicial objects. Suppose that C is a category. By definition,a simplicial object in C (respectively, a cosimplicial object in C) is a functor op ! C (respectively, a functor ! C). Morphisms of (co)simplicial objects arenatural transformations of functors.For a simplicial (respectively, cosimplicial) object F we denote the object F .Œn/ 2C by F n (respectively, F n ).2.2 Cosimplicial vector spaces. Let V be a cosimplicial vector space. We denoteby C .V / the associated complex with component C n .V / D V n and the differential@ n W C n .V / ! C nC1 .V / defined by @ n D P i . 1/i @ n i , where @n iis the map inducedby the i th face map Œn ! Œn C 1. We denote cohomology of this complex by H .V /.The complex C .V / contains a normalized subcomplex xC .V /. Here xC n .V / DfV 2 V n j si nv D 0g, where sn iW Œn ! Œn 1 is the i th degeneracy map. Recall thatthe inclusion xC .V / ! C .V / is a quasiisomorphism.Starting from a cosimplicial vector space V one can construct a new cosimplicialvector space yV as follows. For every W Œn ! set yV D V .n/ . Suppose givenanother simplex W Œm ! and morphism W Œm ! Œn such that D ı , i.e., is a morphism of simplices ! . The morphism .0n/ factors uniquely into 0 !.0/ ! .m/ ! n, which, under , gives the factorization of .0n/W .0/ ! .n/(in ) into.0/ f ! .0/ g ! .m/ ! h .n/; (2.1)where g D .0m/. The map .m/ ! .n/ induces the map W yV ! yV : (2.2)Set now yV n D Q yV . The maps (2.2) endow yV with the structure of a cosimplicialvector space. We then have the following well-knownŒn! result: