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246 U. Bunke, T. Schick, M. Spitzweck, and A. ThomLemma 2.19. In Hom D.ShAb S/.B; AŒ2/ we have the equalityY.K/ D Y 0 .K/:Proof. We consider the morphism of complexes W K B ! K A given by obvious mapsin the diagramX Y B It fits intoBA X Y .ˇ K A ˛AŒ2B ı AŒ2.K BIt suffices to show that the two squares commute. In fact we will show that ishomotopic to ˇ ı and ˛ ı ı. Note that ˇ ı is given byX Y B Aid X00Y .The dotted arrows in this diagram gives the zero homotopy of this difference.Similarly ˛ ı ı is given byX Y B 0A0 XidY ,and we have again indicated the required zero homotopy.2.5.10 The equivalence between the two-categories PIC.S/ and C 0 .S/ (see 2.16) allowsus to classify equivalence classes of Picard stacks with fixed H i .P / Š A i , i D 1; 0,A i 2 Sh Ab S. An equivalence of such Picard stacks is an equivalence which inducesthe identity on the cohomology.Lemma 2.20. The set Ext PIC.S/ .A 0 ;A 1 / of equivalence classes of Picard stacks Pwith given isomorphisms H i .P / Š A i , i D 0; 1, A i 2 Sh Ab S is in bijection withExt 2 Sh Ab S .A 0;A 1 /.Proof. We define a map W Ext PIC.S/ .A 0 ;A 1 / ! Ext 2 Sh Ab S .A 0;A 1 / (12)