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

We define the fundamental 2-crossed complex $\Omega^\infty(X)$ of a reduced CW-complex $X$ from Ellis’ fundamental squared complex $\rho^\infty(X)$ thereby proving that $\Omega^\infty(X)$ 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 $\Omega^\infty(X)$ 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.

Keywords

crossed module, crossed square, squared complex, 2-crossed module, 2-crossed complex, quadratic module, 3-type, Gray enriched category

2010 Mathematics Subject Classification

18D05, 18D20, 55Q05, 55Q15, 55U35

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Published 25 January 2012