Title: Normalisation for the fundamental crossed complex of a simplicial set
Author(s): Ronald Brown, Rafael Sivera
Author's e-mail address: r.brown@bangor.ac.uk, Rafael.Sivera@uv.es
(Optional) Author's mailing address: R. Brown University of Wales, Bangor,
Dean St., Bangor, Gwynedd LL57 1UT, U.K.
R. Sivera, Departamento de Geometria y Topologia, Universitat de Valencia,
46100 Burjassot, Valencia, Spain
(Optional) Included ps or eps files:
(Optional) AMS classification number: 8D10, 18G30, 18G50, 20L05, 55N10, 55N25, 55U10,
55U99
(Optional) Other useful information (such as arXive submission number): math.AT/0611728
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 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, such as
the monoidal closed structure.