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# Homology, Homotopy and Applications

## Volume 13 (2011)

### Number 2

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

Pages: 129 – 157

DOI: http://dx.doi.org/10.4310/HHA.2011.v13.n2.a9

#### Author

#### Abstract

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