  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.}
