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Duality for topological abelian group stacksand T -dualityUlrich Bunke, Thomas Schick, Markus Spitzweck, and Andreas ThomContents1 Introduction ....................................... 2281.1 A sheaf theoretic version of Pontrjagin duality .................. 2281.2 Picard stacks ................................... 2291.3 Duality of Picard stacks .............................. 2301.4 Admissible groups ................................ 2321.5 T -duality ..................................... 2342 Sheaves of Picard categories .............................. 2362.1 Picard categories ................................. 2362.2 Examples of Picard categories .......................... 2382.3 Picard stacks and examples ............................ 2392.4 Additive functors ................................. 2402.5 Representation of Picard stacks by complexes of sheaves of groups ....... 2423 Sheaf theory on big sites of topological spaces ..................... 2483.1 Topological spaces and sites ........................... 2483.2 Sheaves of topological groups .......................... 2483.3 Restriction ..................................... 2513.4 Application to sites of topological spaces ..................... 2613.5 Z mult -modules ................................... 2674 Admissibility of sheaves represented by topological abelian groups .......... 2694.1 Admissible sheaves and groups .......................... 2694.2 A double complex ................................. 2724.3 Discrete groups .................................. 2794.4 Admissibility of the groups R n and T n ..................... 2864.5 Profinite groups .................................. 2894.6 Compact connected abelian groups ........................ 3075 Duality of locally compact group stacks ........................ 3155.1 Pontrjagin Duality ................................. 3155.2 Duality and classification ............................. 3196 T -duality of twisted torus bundles ........................... 3236.1 Pairs and topological T -duality .......................... 3236.2 Torus bundles, torsors and gerbes ......................... 3276.3 Pairs and group stacks .............................. 3326.4 T -duality triples and group stacks ........................ 337References .......................................... 346