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Duality for topological abelian group stacks and T -duality 2291.1.5 We can reformulate Pontrjagin duality in sheaf theoretic terms. The circle groupT belongs to S and gives rise to a sheaf T. Given a sheaf of abelian groups F 2 Sh Ab Swe define its dual byD.F / WD Hom ShAb S .F; T/and observe thatD.G/ Š yGfor an abelian group G 2 S.The image of the natural pairingunder the isomorphismF ˝Z D.F / ! THom ShAb S.F ˝Z D.F /; T/ Š Hom ShAb S.F; Hom ShAb S .D.F /; T//D Hom ShAb S.F; D.D.F ///gives the evaluation mapev F W F ! D.D.F //:Definition 1.2. We call a sheaf of abelian groups F 2 Sh Ab S dualizable, if the evaluationmap ev F W F ! D.D.F // is an isomorphism of sheaves.The sheaf theoretic reformulation of Pontrjagin duality is now:Theorem 1.3 (Sheaf theoretic version of Pontrjagin duality). If G is a locally compactabelian group, then G is dualizable.1.2 Picard stacks1.2.1 A group gives rise to a category BG with one object so that the group appears asthe group of automorphisms of this object. A sheaf-theoretic analog is the notion of agerbe.1.2.2 A set can be identified with a small category which has only identity morphisms.In a similar way a sheaf of sets can be considered as a strict presheaf of categories. Inthe present paper a presheaf of categories on S is a lax contravariant functor S ! Cat.Thus a presheaf F of categories associates to each object A 2 S a category F .A/, andto each morphism f W A ! B a functor f W F.B/ ! F .A/. The adjective lax means,that in addition for each pair of composable morphisms f; g 2 S we have specifiedan isomorphism of functors g ı f f;gŠ .f ı g/ which satisfies higher associativityrelations. The sheaf of categories is called strict if these isomorphisms are identities.1.2.3 A category is called a groupoid if all its morphisms are isomorphisms. A prestackon S is a presheaf of categories on S which takes values in groupoids. A prestack is astack if it satisfies in addition descent conditions on the level of objects and morphisms.For details about stacks we refer to [Vis05].