Gerbes over posets and twisted C*-dynamical systems

Abstract

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 $C^{\ast }$-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 $C^{\ast }$-algebraic level, holonomy leads to a general notion of twisted $C^{\ast }$-dynamical system, based on a generic 2-group instead of the usual adjoint action on the underlying $C^{\ast }$-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.