Gerbes over posets and twisted \(C^*\)-dynamical systems (Q2282976)

Summary: 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 \(\Delta\) 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.