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A (reduced) crossed complex is, intuitively, a non-abelian complex of groups $\ldots \to G_2 \to G_1 \to G_0$ with a $G_0$ action in such that $G_1 \to G_0$ is a crossed module.

There are a couple of constructions that could be called a classifying space of a crossed complex, namely the one given at this MO question, and the simplicial set arising from the $\bar{W}$ functor applied to the simplicial group which is the group T-complex associated to the crossed complex. (Aside: I would be tempted to call this the groupal nerve, as opposed to the simplicial set constructed in the process of forming the classifying space.) Then one can apply geometric realisation to get a space.

My question is this:

Does the usual classifying space functor from crossed complexes to topological spaces lift, up to homotopy, through the functor $|\bar{W} - |: sGrp \to Top$?

Edit: An equivalent formulation is this: for $T:Crs \to sGrp$ the functor from crossed complexes to simplicial groups in the above paragraph (this is half of the relevant Dold-Kan correspondence for crossed complexes), $N:Crs \to sSet$ the nerve functor and $\bar{W}:sGrp \to sSet$ the classifying space functor, do we have a (weak) homotopy equivalence $$\bar{W}T(G) \sim NG?$$

2 fixed typo in title

1

# nerves of crossed comples, group T-complexes and classifying spaces

A (reduced) crossed complex is, intuitively, a non-abelian complex of groups $\ldots \to G_2 \to G_1 \to G_0$ with a $G_0$ action in such that $G_1 \to G_0$ is a crossed module.

There are a couple of constructions that could be called a classifying space of a crossed complex, namely the one given at this MO question, and the simplicial set arising from the $\bar{W}$ functor applied to the simplicial group which is the group T-complex associated to the crossed complex. (Aside: I would be tempted to call this the groupal nerve, as opposed to the simplicial set constructed in the process of forming the classifying space.) Then one can apply geometric realisation to get a space.

My question is this:

Does the usual classifying space functor from crossed complexes to topological spaces lift, up to homotopy, through the functor $|\bar{W} - |: sGrp \to Top$?