TOTAL SPACE OF ABELIAN GERBES

We present a generalization of the construction of a principal G-bundle from a one Čech cocycle to the case of higher abelian gerbes. We prove that the sheaf of local sections of the associated bundle to a higher abelian gerbe is isomorphic to the sheaf of sections of the gerbe itself. Our main result states that equivalence classes of higher abelian gerbes are in bijection with isomorphism classes of the corresponding bundles. We also present topological characterization of those bundles. In the last section, we show that the usual notion of Ehresmann connection leads to the gerbe connection for higher ℂ*-gerbes.