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Duality for topological abelian group stacks and T -duality 255and the map I jY ! J inducesRHom ShAb S .F; G/ jY Š Hom ShAb S=Y .F jY ;I jY /! Hom ShAb S=Y .F jY ;J/Š RHom ShAb S=Y .F jY ;G jY /:(15)If the restriction f W ShS ! ShS=Y would preserve injectives, then I jY ! J wouldbe a homotopy equivalence and the marked map would be a quasi-isomorphism.In the generality of the present paper we do not know whether f preserves injectives.Nevertheless we show that our assumption on S implies that the marked map in(15) is a quasi isomorphism for all sheaves F 2 Sh Ab S.3.3.10 We first show a special case.Lemma 3.13. If F is representable, then the marked map in (15) is a quasi isomorphism.Proof. Let .U ! Y/2 S=Y and .A ! Y/2 S. Then on the one hand we haveHom ShAb S=Y .Z.A/ jY ;I jY /.U ! Y/D Hom .Sh S=Y /=.U !Y/ ..A jY / j.U !Y/ ;.I jY / jU !Y /(by 3.2.7)Š Hom Sh S=U .A jU ;I jU /(by 3.3.7):Since A jU is representable by our assumption on S there exists .B ! U/2 S=U suchthat A jU Š B ! U .Since the restriction functor is exact (Lemma 3.8) and the restriction of an injectivesheaf from S to S=Y is still flabby (Lemma 3.11) we see that G jU ! I jU is a flabbyresolution. It follows thatOn the other handHom Sh S=U .A jU ;I jU / Š Hom Sh S=U .B ! U ;I jU /Š I jU .B ! U/Š R.B ! U; G jU /:Hom ShAb S=Y .Z.A/ jY ; J /.U ! Y/Š Hom .Sh S=Y /=.U !Y/ ..A jY / jU !Y ;J jU !Y /Š Hom Sh S=U .A jU ;J jU /Š Hom Sh S=U .B ! U ;J jU /Š J.B ! U/Š R.B ! U; G jU /: