On Picard 2-Stacks and Length 3 Complexes of Abelian Sheaves

In Seminaire de Geometrie Algebrique 4 (SGA4), Expose XVIII, Pierre Deligne proves that to any Picard stack one can associate a complex of abelian sheaves of length 2. He also studies the morphisms between such stacks and shows that such a morphism defines a class of fractions in the derived category of complexes of abelian sheaves of length 2. From these two preliminary results, he finally deduces that the derived category of complexes of abelian sheaves of length 2 is equivalent to the category of Picard stacks with morphisms being the isomorphism classes. In this dissertation, we generalize his work, following closely his steps in SGA4, to the case of Picard 2-stacks. But this generalization requires first a clear description of a Picard 2-category as well as of a 2-functor between such 2-categories that respects Picard structure. Once this has been done, we can talk about category of Picard 2-stacks and prove that the derived category of complexes of abelian sheaves of length 3 is equivalent to the category of Picard 2-stacks.