Gerbes over posets and twisted C*-dynamical systems

摘要

A base Delta generating the topology of a space M becomes a partially ordered set (poset), when ordered under inclusion of open subsets. Given a precosheaf over A 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 Delta, 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 pi(1) (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 Delta. It is also shown that any section of a DR-presheaf defines a twisted action of pi(1) (M) on a Cuntz algebra.