Classifying space of a crossed complex

Question

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?

Answer 1

It is indeed (weakly equivalent to) the trivial one, in the model structure on crossed complexes.

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 groups 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 of the more familiar situation with Kan complexes.

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 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).

Answer 2

This is a belated reply to this, but I had no time to look at mathoverflow while finishing the book on "Nonabelian algebraic topology:..." .

First it should be emphasised that we get strict fundamental crossed complexes and $\infty$-groupoids (cubical or simplicial $T$-complex) for filtered spaces, not for a "bare" space. Indeed the fact that this strict theory works for filtered spaces, and so generalises the usual fundamental groupoid of a space with a set of base points, is part of the case for filtered spaces as a basic concept in algebraic topology. It will take time for this penny to drop!

The simplicial nerve of a crossed complex gives the usual nerve if the crossed complex is simply a groupoid in dimension 1.

The simplicial nerve of a crossed module is described in dimensions 1,2,3 on p. 323 of that book. See also the last part of section 3 of

`Groupoids and crossed objects in algebraic topology', Homology, homotopy and applications, 1 (1999) 1-78.

for a crossed module $\mu: M \to P$ the nerve $K$ is given by:

$K_0 = {0}, K_1=P$. The 2-simplices of $K$ are quadruples $k=(m;p,q,r)$ such that $m \in M, p,q,r \in P$ and $\mu m = q p r^{-1} $ with $\partial _0 k=p, \partial _1 k=r, \partial _2 k=q$. The 3-simplices of $K$ are quadruples $(k_0,k_1,k_2,k_3)$ of 2-simplices such that if $k_i=(m_i;p_i,q_i,r_i), i=0,\ldots,3$, then \begin{equation} m_0^{p_3} m_2 m_1^{-1} m_3^{-1} =1, \end{equation}and the edges of the 2-simplices $k_i$ match up to form a 3-simplex. For $n\geqslant 4$, an $n$-simplex of $K$ is an $(n+1)$-tuple of $(n-1)$-simplices of $K$, whose faces match up appropriately.

Actually in the reduced (one pointed) case this construction of the nerve of a crossed complex (but called a "group system") is essentially in

Blakers, A. Some relations between homology and homotopy groups. Ann. of Math. (2) 49 (1948) 428--461.