  Gerbes are locally connected presheaves of groupoids. They are classified up
to local weak equivalence by path components in a 2-cocycle category taking
values in all sheaves of groups, their isomorphisms and homotopies. If F is a
full presheaf of sheaves of groups, isomorphisms and homotopies, then [*,BF] is
isomorphic to equivalence classes of gerbes locally equivalent to groups
appearing in F. Giraud's non-abelian cohomology object of equivalence classes
of gerbes with band L is isomorphic to morphisms in the homotopy category from
the point * to the homotopy fibre over L for a map defined on BF and taking
values in the classifying space for the stack completion of the fundamental
groupoid of F.
