Let 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 D(S) of abelian sheaves on S.
The preview is over
If you wish to continue, please create your membership or