Brown defines the classifying space of a crossed complex in the following way.


Given a filtration X\* of a space X, define the fundamental crossed complex by:
C\_0 = X\_0, C\_1=\pi(X\_1,X\_0) (the fundamental groupoid), C\_n = the family of groups \pi(X\_n,X\_n-1,p) for all p in X\_0.


Now let Δ^n be the cell complex of the standard n-simplex, with its skeletal filtration. The crossed complex \pi(π^n) is then written \pi[n]. The nerve NC of a crossed complex C is defined to be the simplicial set given in dimension n by (NC)n = Crs(\pi[n],C), where Crs(-,-) is the internal hom in the category of crossed complexes.


The think I don't understand is that each of the n-simplices are contractible, so why wouldn't the fundamental crossed complex associated to the filtration of Δ^n be trivial?



Classifying space of a crossed complex
It is indeed (weakly equivalent to) the trivial one, in the [model structure on crossed complexes](http://ncatlab.org/nlab/show/folk+model+structure). 


Your question seems to indicate that you think this is a problem. But it is not: each cell of NC is contractible (as it is for each cell of a space!) but there may still be simplicial maps


partial Delta^n --> NC


from the simplicial set that is the boundary of the n-simplex into NC that represent non-trivial elements of [simplicial homotopy group](http://ncatlab.org/nlab/show/simplicial+homotopy+group)s of NC.


To see what is going on, in may be helpful to realize that the situation with crossed complexes is the analogue in the context of [*strict* oo-groupoids](http://ncatlab.org/nlab/show/strict+omega-groupoid) of the more familiar situation with [Kan complex](http://ncatlab.org/nlab/show/Kan+complex)es.


A Kan complex is a model for a weak oo-groupoid. A crossed complex is a strict oo-groupoid (a more restrictive notion).


To any topological space X is associated its singular simplicial Kan complex S(X) = Hom(Delta^bullet,X)-- which we may think of as the [fundamental weak oo-groupoid](http://ncatlab.org/nlab/show/fundamental+infinity-groupoid) of X. This is the weak oo-groupoid version of Ronnie Brown's fundamental crossed complex.


Indeed, the nerve of Brown's fundamental crossed complex gives a Kan complex that approximates the full S(X). 


