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242 U. Bunke, T. Schick, M. Spitzweck, and A. Thom2.4.3 Let now P 1 ;P 2 be two Picard stacks on S.Definition 2.8. An additive functor F W P 1 ! P 2 is a morphism of stacks togetherwith a two-morphismP 1 P C 1 P 1FF F P 2 P C 2 P 2which induces for each U 2 S an additive functor F U W P 1 .U / ! P 2 .U / of Picard categories.An isomorphism between additive functors uW F 1 ! F 2 is a two-isomorphismof morphisms of stacks which induces for each U 2 S an isomorphism of additivefunctors u U W F 1;U ! F 2;U .As in the case of Picard categories, the additive functors between Picard stacks formagain a Picard category.Definition 2.9. We let Hom.P 1 ;P 2 / denote the groupoid of additive functors from P 1 ,P 2 .We get a two-category PIC.S/ of Picard stacks on S.Definition 2.10. We let HOM.P 1 ;P 2 / denote the Picard category of additive functorsbetween Picard stacks P 1 and P 2 .2.4.4 Let P;Q be Picard stacks on the site S. By localization we define a Picard stackHOM.P; Q/.Definition 2.11. The Picard stack HOM.P; Q/ is the sheaf of Picard categories givenbyHOM PIC.S/ .P; Q/.U / WD HOM.P jU ;Q jU /:A priori this describes a prestack, but one easily checks the stack conditions.2.5 Representation of Picard stacks by complexes of sheaves of groups2.5.1 There is an obvious notion of a Picard prestack. Furthermore, there is an associatedPicard stack construction a such that for a Picard prestack P and a Picard stackQ we have a natural equivalenceHom.aP; Q/ Š Hom.P; Q/ (8)2.5.2 Let S be a site. We follow Deligne, SGA 4.3, Expose XVIII, [Del]. Let C.S/ bethe two-category of complexes of sheaves of abelian groupsK W 0 ! K 1 d ! K 0 ! 0which live in degrees 1; 0. Morphisms in C.S/ are morphisms of complexes, andtwo-isomorphisms are homotopies between morphisms.To such a complex we associate a Picard stack ch.K/ on S as follows. We firstdefine a Picard prestack p ch.K/.