# Gerbes over posets and twisted C*-dynamical systems

### Ezio Vasselli

Università di Roma II - Tor Vergata, Rome, Italy

## Abstract

A base $Δ$ generating the topology of a space $M$ becomes a partially ordered set (poset), when ordered under inclusion of open subsets. Given a precosheaf over $Δ$ 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 $Δ$, 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 $π_{1}(M)$. 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 $Δ$. It is also shown that any section of a DR-presheaf defines a twisted action of $π_{1}(M)$ 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. 1151–1208

DOI 10.4171/JNCG/347