process

230 U. Bunke, T. Schick, M. Spitzweck, and A. Thom1.2.4 A sheaf of groups F 2 Sh Ab S gives rise to a prestack p BF which associates toU 2 S the groupoid p BF.U/ WD BF .U /. This prestack is not a stack in general. Butit can be stackified (this is similar to sheafification) to the stack BF .1.2.5 For an abelian group G the category BG is actually a group object in Cat. Thegroup operation is implemented by the functor BG BG ! BG which is obvious onthe level of objects and given by the group structure of G on the level of morphisms.In this case associativity and commutativity is strictly satisfied. In a similar manner,for F 2 Sh Ab S the prestack p BF on S becomes a group object in prestacks on S.1.2.6 A group G can of course also be viewed as a category G with only identitymorphisms. This category is again a strict group object in Cat. The functor G G ! Gis given by the group operation on the level of objects, and in the obvious way on thelevel of morphisms.1.2.7 In general, in order to define group objects in two-categories like Cat or the categoryof stacks on S, one would relax the strictness of associativity and commutativity.For our purpose the appropriate relaxed notion is that of a Picard category which wewill explain in detail in 2. The corresponding sheafified notion is that of a Picard stack.We let PIC S denote the two-category of Picard stacks on S.1.2.8 A sheaf of groups F 2 S Ab S gives rise to a Picard stack in two ways.First of all it determines the Picard stack F which associates to U 2 S the Picardcategory F.U/ in the sense of 1.2.6. The other possibility is the Picard stack BFobtained as the stackification of the Picard prestack p BF .1.2.9 Let A 2 PIC S be a Picard stack. The presheaf of its isomorphism classes ofobjects generates a sheaf of abelian groups which will be denoted by H 0 .A/ 2 Sh Ab S.The Picard stack A gives furthermore rise to the sheaf of abelian groups H 1 .A/ 2Sh Ab S of automorphisms of the unit object.A Picard stack A 2 PIC S fits into an extension (see 2.5.12)BH 1 .A/ ! A ! H 0 .A/: (1)1.3 Duality of Picard stacks1.3.1 It is an essential observation by Deligne that the two-category of Picard stacksPIC S admits an interior HOM PIC S . Thus for Picard stacks A; B 2 PIC S we have aPicard stackHOM PIC S .A; B/ 2 PICSof additive morphisms from A to B (see 2.4).