Extensions of picard 2-stacks and the cohomology groups Exti of length 3 complexes

The aim of this paper is to define and study the 3-category of extensions of Picard 2-stacks over a site S and to furnish a geometrical description of the cohomology groups Exti of length 3 complexes of abelian sheaves. More precisely, our main theorem furnishes (1) a parametrization of the equivalence classes of objects, 1-arrows, 2-arrows, and 3-arrows of the 3-category of extensions of Picard 2-stacks by the cohomology groups Exti, and (2) a geometrical description of the cohomology groups Exti of length 3 complexes of abelian sheaves via extensions of Picard 2-stacks. To this end, we use the triequivalence between the 3-category 2Picard(S) of Picard 2-stacks and the tricategory T[−2,0] (S) of length 3 complexes of abelian sheaves over S introduced by the second author in [12], and we define the notion of extension in this tricategory T[−2,0] (S), getting a pure algebraic analog of the 3-category of extensions of Picard 2-stacks. The calculus of fractions that we use to define extensions in the tricategory T[−2,0] (S) plays a central role in the proof of our main theorem.