Biextensions of Picard stacks and their homological interpretation

Let S be a site. We introduce the 2-category of biextensions of strictly commutative Picard S-stacks. We define the pull-back, the push-down, and the sum of such biextensions and we compute their homological interpretation: if P,Q and G are strictly commutative Picard S-stacks, the equivalence classes of biextensions of (P,Q) by G are parametrized by the cohomology group Ext1([P]⊗L[Q],[G]), the isomorphism classes of arrows from such a biextension to itself are parametrized by the cohomology group Ext0([P]⊗L[Q],[G]) and the automorphisms of an arrow from such a biextension to itself are parametrized by the cohomology group Ext−1([P]⊗L[Q],[G]), where [P],[Q] and [G] are the complexes associated to P,Q and G respectively.