  The bigerbes introduced here give a refinement of the notion of 2-gerbes,
representing degree four integral cohomology classes of a space. Defined in
terms of bisimplicial line bundles, bigerbes have a symmetry with respect to
which they form 'bundle 2-gerbes' in two ways; this structure replaces higher
associativity conditions. We provide natural examples, including a
Brylinski-McLaughlin bigerbe associated to a principal G-bundle for a simply
connected simple Lie group. This represents the first Pontryagin class of the
bundle, and is the obstruction to the lifting problem on the associated
principal bundle over the loop space to the structure group consisting of a
central extension of the loop group; in particular, trivializations of this
bigerbe for a spin manifold are in bijection with string structures on the
original manifold. Other natural examples represent 'decomposable' 4-classes
arising as cup products, a universal bigerbe on K(Z,4) involving its based
double loop space, and the representation of any 4-class on a space by a
bigerbe involving its free double loop space. The generalization to
'multigerbes' of arbitrary degree is also described.
