Homology, Homotopy and Applications

Volume 13 (2011)

Number 2

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

Pages: 129 – 157

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


João Faria Martins (Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, Caparica, Portugal)


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.


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

Published 25 January 2012