The fundamental 2-crossed complex of a reduced cw-complex

We define the fundamental 2-crossed complex Ω∞ (X) of areduced CW-complex X from Ellis’ fundamental squared com-plex ρ∞ (X) thereby proving that Ω∞ (X) is totally free onthe set of cells of X. This fundamental 2-crossed complex hasvery good properties with regard to the geometrical realisa-tion of 2-crossed complex morphisms. After carefully discussingthe homotopy theory of totally free 2-crossed complexes, weuse Ω∞ (X) to give a new proof that the homotopy categoryof pointed 3-types is equivalent to the homotopy category of2-crossed modules of groups. We obtain very similar results tothe ones given by Baues in the similar context of quadraticmodules and quadratic chain complexes.