process

Duality for topological abelian group stacks and T -duality 239as the map ŒŒq; p; r ! Œq; Œp; r. Finally we let W Q ˝G P ! P ˝G Q be givenby Œq; p ! Œp; q.We claim that .˝G;;/is a strict Picard category. Let I be a collection of objectsof BG.B/. We choose an ordering of I . Then we can write each element of j 2 N.I/in a unique way as ordered product f D P 1 P 2 :::P r with P 1 P 2 P r . Wedefine F .f / WD P 1˝G .P 2˝G .˝G P r / /. We let a P W F .P / ! P , P 2 I ,bethe identity. Furthermore, for f; g 2 N.I/ we define a f;g W F .fg/ ! F .f / ˝B F .g/as the map induced the permutation which puts the factors of f and g in order (and sothat the order to repeated factors is not changed). Commutativity of (6) is now clear.In order to check the commutativity of (7) observe that W Q ˝G Q ! Q ˝G Q is theidentity, as Œqg; q 7! Œq; qg D Œqg; q.2.3 Picard stacks and examples2.3.1 We consider a site S. A prestack on S is a lax associative functor P W S op !groupoids, where from now on by a groupoid we understand an essentially small category3 in which all morphisms are isomorphisms. Lax associative means that as a part ofthe data for each composable pair of maps f; g in S there is an isomorphism of functorsl f;g W P.f/ı P.g/ ! P.g ı f/, and these isomorphisms satisfy higher associativityrelation. A prestack is a stack if it satisfies the usual descent conditions one the levelof objects and morphisms. For a reference of the language of stacks see e.g. [Vis05].Definition 2.4. A Picard stack P on S is a stack P together with an operationCW P P ! P and transformations , which induce for each U 2 S the structureof a Picard category on P.U/.2.3.2 Let Sh Ab S denote the category of sheaves of abelian groups on S. Extending theexample 2.2.1 to sheaves we can view each object F 2 Sh Ab S as a Picard stack, whichwe will again denote by F .2.3.3 We can also extend Example 2.2.2 to sheaves. In this way every sheaf F 2 Sh Ab Sgives rise to a Picard stack BF .2.3.4 We now extend the example 2.2.3 to sheaves. First of all note that one can definea Picard category EXT.G ; H/ of extensions of sheaves 0 ! H ! E ! G ! 0 as adirect generalization of 2.2.3. Then we define a prestack EXT.G ; H/ which associatesto U 2 S the Picard category EXT.G ; H/.U / WD EXT.G jU ; H jU /. One checks thatEXT.G ; H/ is a stack.2.3.5 Let G 2 Sh Ab S be a sheaf of abelian groups. We consider G as a group object inthe category ShS of sheaves of sets on S. AG -torsor is an object T 2 ShS with an actionof G such that T is locally isomorphic to G . We consider the category of G -torsorsTors.G / and their isomorphisms. We define the functor CW Tors.G / Tors.G / !3 Later, e.g. in 2.5.3, we need that the isomorphism classes in a groupoid form a set.