Gerbes over posets and twisted C*-dynamical systems

Abstract

A base ∆ generating the topology of a space M becomes a partially ordered set (poset), when ordered under inclusion of open subsets. Given a precosheaf over ∆ of fixed-point spaces (typically C * -algebras) under the action of a group G , in general one cannot find a precosheaf of G -spaces having it as fixed-point precosheaf. Rather one gets a gerbe over ∆ , that is, a "twisted precosheaf" whose twisting is encoded by a cocycle with coefficients in a suitable 2-group. We give a notion of holonomy for a gerbe, in terms of a non-abelian cocycle over the fundamental group π1(M ) . At the C * -algebraic level, holonomy leads to a general notion of twisted C * -dynamical system, based on a generic 2-group instead of the usual adjoint action on the underlying C * -algebra. As an application of these notions, we study presheaves of group duals (DR-presheaves) and prove that the dual object of a DR-presheaf is a group gerbe over ∆ . It is also shown that any section of a DR-presheaf defines a twisted action of π1(M ) on a Cuntz algebra.