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244 U. Bunke, T. Schick, M. Spitzweck, and A. Thom2.5.5 We consider two morphisms f; g W K ! L of complexes in C.S/. A homotopyH W f ! g is a morphism of sheaves H W K 0 ! L 1 such that f 0 g 0 D d L ı Hand f 1 g 1 D H ı d K . It is easy to see that H induces an isomorphism of additivefunctors p ch.H /W p ch.f / ! p ch.g/ and thereforech.H /W ch.f / ! ch.g/by the associated stack construction and (8).One can show that the isomorphisms p ch.f / ! p ch.g/ correspond precisely tohomotopies H W f ! g. This implies that the morphisms ch.f / ! ch.g/ alsocorrespond bijectively to these homotopies.2.5.6 We have the following result.Lemma 2.13 ([Del, 1.4.13]). 1. For every Picard stack P there exists a complexK 2 C.S/ such that P Š ch.K/.2. For every additive functor F W ch.K/ ! ch.L/ there exists a quasi isomorphismk W K 0 ! K and a morphism l W K 0 ! L in C.S/ such that F Š ch.l/ ı ch.k/ 1 .2.5.7 Let PIC [ .S/ be the category of Picard stacks obtained from the two-categoryPIC S by identifying isomorphic additive functors.Proposition 2.14 ([Del, 1.4.15]). The construction ch gives an equivalence of categories1;0 .Sh Ab S/ ! PIC [ .S/;D Œwhere D Œ 1;0 .Sh Ab S/ is the full subcategory of D C .Sh Ab S/ of objects whose cohomologyis trivial in degrees 62 ¹ 1; 0º.Lemma 2.15 ([Del, 1.4.16]). Let K; L 2 C.S/ and assume that L 1 is injective.1. p ch.L/ is already a stack.2. For every morphism F 2 Hom PIC.S/ .ch.K/; ch.L// there exists a morphismf 2 Hom C.ShAb S/.K; L/ such that ch.f / Š F .Let C 0 .S/ C.S/ denote the full sub-two-category of complexes 0 ! K 1 !K 0 ! 0 with K 1 injective.Lemma 2.16 ([Del, 1.4.17]). The construction ch induces an equivalence of the twocategoriesPIC.S/ and C 0 .S/.2.5.8 Finally we give a characterization of the Picard stack Hom.ch.K/; ch.L//.Lemma 2.17 ([Del, 1.4.18.1]). Assume that L 1 is injective. Then we have an equivalencech. 0 Hom ShAb S .K; L// ! HOM PIC.S/ .ch.K/; ch.L//: