The Host-Kra-Ziegler factors of an ergodic system are higher order analogues of the classical Kronecker factor. An important fact is that they are characteristic for various non-conventional ergodic averages including the one used by Furstenberg in order to establish Szemeredi's theorem. A key structural result implies that these factors are (uniquely ergodic) inverse limits of nilsystems - in particular (just as in the Kronecker case) belong to topological category. In a work in progress we propose a new way of "passing from the measurable to the topological" which has the advantage of working for arbitrary countable Abelian ergodic actions. The main tools are the Camarena-Szegedy concept of cocycle and a generalization of the Host-Kra-Maass structure theorem in a recent joint work with Freddie Manners and Peter Varju. As an application one finds characteristic factors for various non-conventional ergodic averages in the context of countable Abelian ergodic actions.