  Let S be a site. First we define the 3-category of torsors under a Picard
S-2-stack and we compute its homotopy groups. Using calculus of fractions we
define also a pure algebraic analogue of the 3-category of torsors under a
Picard S-2-stack. Then we describe extensions of Picard S-2-stacks as torsors
endowed with a group law on the fibers. As a consequence of such a description,
we show that any Picard S-2-stack admits a canonical free partial left
resolution that we compute explicitly. Moreover we get an explicit right
resolution of the 3-category of extensions of Picard S-2-stacks in terms of
3-categories of torsors. Using the homological interpretation of Picard
S-2-stacks, we rewrite this three categorical dimensions higher right
resolution in the derived category of abelian sheaves on S.
