Bigerbes

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