Smaller L_∞-model for the cover of the classifying space up to homotopy is imminent. This model retains all the information of the larger models found in [32, 33, 38]. We give immediate consequences of -structure transferred interacting with one of the grading and study structural properties of Koszul spaces deciding the model.

classifying spaces for groups and H-spaces, homology of classifying spaces, characteristics, L_∞-structure transferred, Koszul space.