process

Duality for topological abelian group stacks and T -duality 331Observe that the group of automorphisms of every object in RH E .T / is isomorphicto Map.T; K/ ı (the superscript ı indicates that we take the underlying set). For a mapuW T ! T 0 over B there is a natural pull-back functor RH E .u/W RE H .T 0 / ! RH E .T /.Let S be a site as in 3.1.2 and assume that K; G; H; B belong to S. In this case it iseasy to check thatT 7! RH E .T /is a gerbe with band K jB on S=B.6.2.11 Let W E ! B be a G-principal bundle. Note that the pull-back E ! E hasa canonical section and is therefore trivialized. A trivialized G-bundle has a canonicalH -reduction. In other words, there is a canonical map of stacks over BcanW E ! R E H : (54)Note that an object of E.T / is a map T ! E; to this we assign the pull-back of thecanonical H -reduction of E.6.2.12 The construction Prin B .G/ 3 E 7! RH E 2 Gerbe.K/ is functorial in E andthus induces a natural map of sets of isomorphism classesr W H 0 .Prin B .G// ! H 0 .Gerbe.K//:Lemma 6.15. If H ! G is surjective and has local sections, and H 1 .B; H / ŠH 2 .B; H / Š 0, thenis a bijection.r W H 0 .Prin B .G// ! H 0 .Gerbe.K//Proof. The exact sequence 0 ! K ! H ! G ! 0 induces by 3.4 an exact sequence0 ! K ! H ! G ! 0of sheaves. We consider the following segment of the associated long exact sequencein cohomology:!H 1 .BI H / ! H 1 .BI G/ ı ! H 2 .BI K/ ! H 2 .BI H / ! :By our assumptions ı W H 1 .BI G/ ! H 2 .BI K/ is an isomorphism. One can checkthat the following diagram commutes:H 0 .Prin B .G//rH 0 .Gerbe.K jB //H 1 .BI G/ıd H 2 .BI K/.(55)This implies the result since the vertical maps are isomorphisms.