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Duality for topological abelian group stacks and T -duality 285There is a natural morphism F.j/ ! M j F for all F 2 C.Ab/ J op . The matching spacecondition asserts that a map F ! G in C.Ab/ J op is a fibration if and only ifF.j/ ! G.j / Mj G M j Fis a fibration in C.Ab/, i.e. a level-wise surjection, for all j 2 J . In particular, F isfibrant if F.j/ ! M j F is a level-wise surjection for all j 2 J .For X 2 C.Ab/ J op the fibrant replacement X ! RX induces the morphismlim X ! lim RX Š R lim X:Let now again A denote the identity functor AW J ! Ab for the discrete abeliangroup D.Lemma 4.26. 1. For all U 2 S which are acyclic the diagram of abelian groupsHom ShAb S.A; T/.U / 2 C.Ab/ J op is fibrant.2.lim Hom ShAb S lc-acyc.A; T/ ! R lim Hom ShAb S lc-acyc.A; T/ 2 C.Ab/ J opis a quasi-isomorphism.Proof. The assertion 1. for locally compact acyclic U verifies the assumption of Lemma4.23. Hence 2. follows from 1. We now concentrate on 1. We must show thatis surjective. We haveHom ShAb S .A.j /; T/.U / ! M j Hom ShAb S .A; T/.U / (29)M j Hom ShAb S .A; T/ Š lim Hom Sh Ab S .A jJ j; T/ Š Hom ShAb S .colim A jJ j; T/:The map (29) is induced by the mapcolimA jJj ! F .j /: (30)By Lemma 3.3 we have colim A jJj Š colim A jJj . The map colim A jJj ! A.j / isof course an injection.We finish the argument by the following observation. Let H ! G be an injectivemap of finitely generated groups (we apply this with H WD colimA jJj and G WD F.j/).Then for U 2 Sis a surjection. In fact we haveHom ShAb S .G; T/.U / ! Hom Sh Ab S .H ; T/.U /Hom ShAb S .G; T/.U / Š Hom top-Ab.G; T/.U / (by Lemma 3.5)Š yG.U /