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Deformations of gerbes on smooth manifolds 3532.3 Covers. A cover (open cover) of a space X is a collection U of open subsets ofX such that S U 2U U D X.2.3.1 The nerve of a cover. Let N 0 U D `U 2UU . There is a canonical augmentationmap`U 2U 0 W N 0 U.U ,!X/ ! X:LetN p U D N 0 U X X N 0 Ube the .p C 1/-fold fiber product.The assignment N UW 3 Œp 7! N p U extends to a simplicial space called thenerve of the cover U. The effect of the face map @ n i(respectively, the degeneracymap si n) will be denoted by d i D dn i (respectively, & i D &i n ) and is given by theprojection along the i th factor (respectively, the diagonal embedding on the i th factor).Therefore for every morphism f W Œp ! Œq in we have a morphism N q U ! N p Uwhich we denote by f . We will denote by f the operation .f / of pull-back alongf ;ifF is a sheaf on N p U then f F is a sheaf on N q U.For 0 i n C 1 we denote by pr n i W N nU ! N 0 U the projection onto the i thfactor. For 0 j m, 0 i j n the map pr i0pr imW N n U ! .N 0 U/ mcan be factored uniquely as a composition of a map N n U ! N m U and the canonicalimbedding N m U ! .N 0 U/ m . We denote this map N n U ! N m U by pr n i 0 :::i m.The augmentation map 0 extends to a morphism W N U ! X where the latter isregarded as a constant simplicial space. Its component of degree n n W N n U ! X isgiven by the formula n D 0 ı pr n i. Here 0 i n C 1 is arbitrary.2.3.2 Čech complex. Let F be a sheaf of abelian groups on X. One defines a cosimplicialgroup LC .U; F / D .NUI F /, with the cosimplicial structure inducedby the simplicial structure of N U. The associated complex is the Čech complex ofthe cover U with coefficients in F . The differential L @ in this complex is given byP . 1/ i .d i / .2.3.3 Refinement. Suppose that U and V are two covers of X. A morphism of covers W U ! V is a map of sets W U ! V with the property U .U / for all U 2 U.A morphism W U ! V induces the map NW N U ! N V of simplicial spaceswhich commutes with respective augmentations to X. The map N 0 is determined bythe commutativity ofU N 0 UN 0 .U / N 0 V.It is clear that the map N 0 commutes with the respective augmentations (i.e. is a map ofspaces over X) and, consequently induces maps N n D .N 0 / X nC1 which commutewith all structure maps.