Strictly ergodic distal models and a new approach to the Host-Kra factors

Cocycles are a key object in Antolín Camarena and Szegedy's theory of nilspaces. We introduce variants these cocycles, named nilcycles, enabling us to give conditions which guarantee that an ergodic group extension strictly distal system admits topological model. In particular we show if the base space is dynamical nilspace then model may be chosen for extension. This approach combined with structure theorem Gutman, Manners Varjú applied extensions between successive Host-Kra characteristic factors gives new proof inverse limit nilsystems Zk-actions.