A base generating the topology of a space becomes a partially ordered set (poset), when ordered under inclusion of open subsets. Given a precosheaf over of fixed-point spaces (typically -algebras) under the action of a group , in general one cannot find a precosheaf of -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 . At the -algebraic level, holonomy leads to a general notion of twisted -dynamical system, based on a generic 2-group instead of the usual adjoint action on the underlying -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 on a Cuntz algebra.
Cite this article
Ezio Vasselli, Gerbes over posets and twisted C*-dynamical systems. J. Noncommut. Geom. 13 (2019), no. 3, pp. 1151–1208DOI 10.4171/JNCG/347