  Weak morphisms of non-abelian complexes of length 2, or crossed modules, are
morphisms of the associated 2-group stacks, or gr-stacks. We present a full
description of the weak morphisms in terms of diagrams we call butterflies. We
give a complete description of the resulting bicategory of crossed modules,
which we show is fibered and biequivalent to the 2-stack of 2-group stacks. As
a consequence we obtain a complete characterization of the non-abelian derived
category of complexes of length 2. Deligne's analogous theorem in the case of
Picard stacks and abelian sheaves becomes an immediate corollary. Commutativity
laws on 2-group stacks are also analyzed in terms of butterflies, yielding new
characterizations of braided, symmetric, and Picard 2-group stacks.
Furthermore, the description of a weak morphism in terms of the corresponding
butterfly diagram allows us to obtain a long exact sequence in non-abelian
cohomology, removing a preexisting fibration condition on the coefficients
short exact sequence.
