We define the fundamental 2-crossed complex Ω∞(X) of a reduced CW-complex X from Ellis' fundamental squared complex ρ∞(X) thereby proving that Ω∞(X) is totally free on the set of cells of X. This fundamental 2-crossed complex has very good properties with regard to the geometrical realisation of 2-crossed complex morphisms. After carefully discussing the homotopy theory of totally free 2-crossed complexes, we use Ω∞(X) to give a new proof that the homotopy category of pointed 3-types is equivalent to the homotopy category of 2-crossed modules of groups. We obtain very similar results to the ones given by Baues in the similar context of quadratic modules and quadratic chain complexes.
Homology, Homotopy and Applications, Vol. 13 (2011), No. 2, pp.129-157.
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