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Duality for topological abelian group stacks and T -duality 3155 Duality of locally compact group stacks5.1 Pontrjagin Duality5.1.1 In this subsection we extend Pontrjagin duality for locally compact groups toabelian group stacks whose sheaves of objects and automorphisms are represented bylocally compact groups. In algebraic geometry a parallel theory has been consideredin [DP].The site S denotes the site of compactly generated spaces as in 3.1.2 or one of itssub-sites S lc , S lc-acyc . The reason for considering these sub-sites lies in the fact thatcertain topological groups are only admissible on these sub-sites (see the Definitions4.1, 4.2 and Theorem 4.8).5.1.2 Let F 2 Sh Ab S.Definition 5.1. We define the dual sheaf of F byD.F / WD Hom ShAb S .F; T/:Definition 5.2. We call F dualizable, if the canonical evaluation morphismis an isomorphism of sheaves.c W F ! D.D.F // (44)Lemma 5.3. If G is a locally compact group which together with its Pontrjagin dualHom top-Ab .G; T/ is contained in S, then G 2 Sh Ab S is dualizable.Proof. By Lemma 3.5 we have isomorphismsandHom ShAb S .G; T/ Š Hom top-Ab.G; T/Hom ShAb S .Hom Sh Ab S .G; T/; T/ Š Hom Sh Ab S .Hom top-Ab.G; T/; T/Š Hom top-Ab .Hom top-Ab .G; T/; T/:The morphism c in (44) is induced by the evaluation mapG ! Hom top-Ab .Hom top-Ab .G; T/; T/which is an isomorphism by the classical Pontrjagin duality of locally compact abeliangroups [Fol95], [HM98].5.1.3 The sheaf of abelian groups T 2 Sh Ab S gives rise to a Picard stack BT 2 PIC.S/as explained in 2.3.3. Recall from 2.5.11 the following alternative description of BT.Let TŒ1 be the complex with the only non-trivial entry TŒ1 1 WD T. Then we haveBT Š ch.TŒ1/ in the notation of 2.12.