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228 U. Bunke, T. Schick, M. Spitzweck, and A. Thom1 Introduction1.1 A sheaf theoretic version of Pontrjagin duality1.1.1 A character of an abelian topological group G is a continuous homomorphismW G ! Tfrom G to the circle group T. The set of all characters of G will be denoted by yG. Itis again a group under point-wise multiplication.Assume that G is locally compact. The compact-open topology on the space ofcontinuous maps Map.G; T/ induces a compactly generated topology on this space ofmaps, and hence a topology on its subset yG Map.G; T/. The group yG equippedwith this topology is called the dual group of G.1.1.2 An element g 2 G gives rise to a character ev.g/ 2 yG defined by ev.g/./ WD.g/ for 2 yG. In this way we get a continuous homomorphismThe main assertion of Pontrjagin duality isev W G ! yG:Theorem 1.1 (Pontrjagin duality). If G is a locally compact abelian group, thenev W G ! yG is an isomorphism of topological groups.Proofs of this theorem can be found e.g. in [Fol95] or [HM98].1.1.3 In the present paper we use the language of sheaves in order to encode thetopology of spaces and groups. Let X; B be topological spaces. The space X givesrise to a sheaf of sets X on B which associates to an open subset U B the setX.U / D C.U;X/ of continuous maps from U to X. If G is a topological abeliangroup, then G is a sheaf of abelian groups on B.The space X or the group G is not completely determined by the sheaf it generatesover B. As an extreme example take B D ¹º. Then X and the underlying discretespace X ı induce isomorphic sheaves on B.For another example, assume that B is totally disconnected. Then the sheavesgenerated by the spaces Œ0; 1 and ¹º are isomorphic.1.1.4 But one can do better and consider sheaves which are defined on all topologicalspaces or at least on a sufficiently big subcategory S TOP. We turn S into aGrothendieck site determined by the pre-topology of open coverings (for details aboutthe choice of S see Section 3). We will see that the topology in S is sub-canonical sothat every object X 2 S represents a sheaf X 2 ShS. By the Yoneda Lemma the spaceX can be recovered from the sheaf X 2 ShS represented by X. The evaluation of thesheaf X on A 2 S is defined by X.A/ WD Hom S .A; X/. Our site S will in particularcontain all locally compact spaces.