process

240 U. Bunke, T. Schick, M. Spitzweck, and A. ThomTors.G / by T 1 C T 2 WD T 1 T 2 =G , where we take the quotient by the anti-diagonalaction. The structure of a G -torsor is induced by the action of G on the second factor T 2 .We let and be induced by the associativity transformation of the cartesian productand the flip, respectively. With these structures Tors.G / becomes a Picard category.We define a Picard stack T ors.G / by localization, i.e. we set T ors.G /.U / WDTors.G jU /.We have a canonical map of Picard stacksU W EXT.Z; G / ! T ors.G /which maps the extension E W 0 ! G ! E ! Z ! 0 to the G-torsor U.E/ D 1 .1/.Here Z denotes the constant sheaf on S with value Z. One can check that U is anequivalence of Picard stacks (see 2.4.3 for precise definitions).2.3.6 The example 2.2.4 has a sheaf theoretic interpretation. We assume that S is somesmall subcategory of the category of topological spaces which is closed under takingopen subspaces, and such that the topology is induced by the coverings of spaces byfamilies of open subsets.Let G be a topological abelian group. We have a Picard prestack p BG on Swhich associates to each space B 2 S the Picard category p BG.B/ of G-principalbundles on B. As in 2.2.4 the monoidal structure on p BG.B/ is given by the tensorproduct of G-principal bundles (this uses that G is abelian). We now define BG as thestackification of p BG.The topological group G gives rise to a sheaf G 2 Sh Ab S which associates to U 2 Sthe abelian group G.U / WD C.U;G/. We have a canonical transformation W BG ! T ors.G/:It is induced by a transformation p W p BG ! T ors.G/. We describe the functorp B W p BG.B/ ! T ors.G/.B/ for all B 2 S. Let E ! B be a G-principal bundle.Then we define p B .E/ 2 Tors.G jB / to be the sheaf which associates to each. W U ! B/ 2 S=B the set of sections of E ! U .One can check that is an equivalence of Picard stacks (see 2.4.3 for precisedefinitions).2.4 Additive functors2.4.1 We shall now discuss the notion of an additive functor between Picard stacksF W P 1 ! P 2 .Definition 2.5. An additive functor between Picard categories is a functor F W P 1 ! P 2and a natural transformation F.xCy/ ! F.x/CF.y/such that the following diagramscommute: