process

Duality for topological abelian group stacks and T -duality 3296.2.6 Consider an extension W as in (51) and apply Ext ShAb S.Z;:::/. We get thefollowing piece of the long exact sequence!Hom ShAb S.Z;W/! Hom ShAb S.Z; Z/ı W! Ext 1 Sh Ab S .Z;F/! Ext1 Sh Ab S .Z;W/! :Let 1 2 Hom ShAb S.Z; Z/ be the identity and sete.W/ WD ı W .1/ 2 Ext 1 Sh Ab S .Z;F/:Recall that H 0 .EXT.Z;F// denotes the set of isomorphism classes of the categoryEXT.Z;F/. The following lemma is well-known (see e.g. [Yon60]).Lemma 6.11. The mapinduces a bijectionEXT.Z;F/3 W 7! e.W/ 2 Ext 1 Sh Ab S .Z;F/e W H 0 .EXT.Z;F// ! Ext 1 Sh Ab S .Z;F/:In view of 6.2.5 we have natural bijectionsH 0 .Tors.F // Š U H 0 .EXT.Z;F// Š e Ext 1 Sh Ab S .Z;F/: (52)6.2.7 Let G be an abelian topological group and consider a principal G-bundle Eover B with underlying map W E ! B. By T .E/ WD E ! B 2 ShS=B (see3.2.3) we denote its sheaf of sections. The right action of G on E induces an actionT .E/ G jB ! T .E/.Lemma 6.12. T .E/ is a G jB -torsor.Proof. This follows from the following fact. If X ! B and Y ! B are two maps,thenX B Y ! B Š X ! B Y ! Bin ShS=B. We apply this to the isomorphismE B .B G/ Š E G Š E B Eof spaces over B in order to get the isomorphismT .E/ G jB! T .E/ T .E/: