Normalisation for the fundamental crossed complex of a simplicial set

TL;DR

This paper develops a normalization theorem for the fundamental crossed complex of a simplicial set, providing an algebraic approach to modeling simplices and proving the Homotopy Addition Lemma.

Contribution

It introduces a normalization theorem for the fundamental crossed complex, extending classical results to a more complex algebraic structure associated with simplicial sets.

Findings

Established a normalisation theorem for the fundamental crossed complex

Provided an algebraic proof of the Homotopy Addition Lemma

Extended classical normalization results to crossed complexes

Abstract

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.}