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Duality for topological abelian group stacks and T -duality 2431. For U 2 S we define the set of objects of p ch.K/.U / as K 0 .U /.2. For x;y 2 K 0 .U / we define the set of morphisms byHomp ch.K/.U /.x; y/ WD ¹f 2 K 1 .U / j df D x3. The composition of morphisms is addition in K 1 .U /.4. The functor CW p ch.K/.U / p ch.K/.U / ! p ch.K/.U / is given by addition ofobjects and morphisms. The associativity and the commutativity constraints arethe identities.Definition 2.12. We define ch.K/ as the Picard stack associated to the prestack p ch.K/,i.e. ch.K/ WD a p ch.K/.2.5.3 If P is a Picard stack, then we can define a presheaf p H 0 .P / which associatesto U 2 S the group of isomorphism classes of P.U/. We let H 0 .P / be the associatedsheaf. Furthermore we have a sheaf H 1 .P / which associates to U 2 S the group ofautomorphisms Aut.e U /, where e U 2 P.U/is a choice of a neutral object with respectto C. Since the neutral object e U is determined up to unique isomorphism, the groupAut.e U / is also determined up to unique isomorphism. Note that Aut.e U / is abelian, asshows the following diagramyº:e C eidCf e C egCid e C eŠ f g e e eand the fact that .id C f/ı .g C id/ D g C f D .g C id/ ı .id C f/.If V ! U is a morphism in S, then we have a unique isomorphism f W .e U / jV ! eVwhich induces a group homomorphism f ııf 1 W Aut.e U / jV ! Aut.e V /. Thishomomorphism is the structure map of the sheaf H 1 .P / which is determined up tounique isomorphism in this way.We now observe thatH 1 .ch.K// Š H 1 .K/; H 0 .ch.K// Š H 0 .K/: (9)2.5.4 A morphism of complexes f W K ! L in C.S/, i.e. a diagramŠŠ0 K1d K K0 0f 1d0 L1 L L0 0,induces an additive functor of Picard prestacks p ch.f /W p ch.K/ ! p ch.L/ and, by thefunctoriality of the associated stack construction, an additive functorch.f /W ch.K/ ! ch.L/:In view of (9), it is an equivalence if and only if f is a quasi isomorphism.f 0