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Normalisation for the fundamental crossed complex of a simplicial set

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Ronald Brown and Rafael Sivera

Crossed complexes are shown to have an algebra
sufficiently rich to model the geometric inductive definition of
simplices, and so to give a purely algebraic proof of the Homotopy
Addition Lemma (HAL) for the boundary of a simplex. This leads to
the {\it fundamental crossed complex} of a simplicial set. The main
result is a normalisation theorem for this fundamental crossed
complex, analogous to the usual theorem for simplicial abelian
groups, but more complicated to set up and prove, because of the
complications of the HAL and of the notion of homotopies for crossed
complexes. We start with some historical background, and give a
survey of the required basic facts on crossed complexes.

Journal of Homotopy and Related Structures, Vol. 2(2007), No. 2, pp. 49-79