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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 $$ \bar{W}T(G) \sim NG? $$

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Yes, that's true. –  Fernando Muro May 26 '11 at 6:22
    
Ok - more details? Is it written down anywhere? This question came up in conversation with Andre Joyal, but I got stuck on this bit. My email is first.last@adelaide.e d u.a u if you would like it. –  David Roberts May 26 '11 at 22:31

1 Answer 1

Perhaps the following is related to your question:

Up to isomorphy, $\overline{\mathrm{W}}\colon \mathbf{sGrp} \rightarrow \mathbf{sSet}$ is the composite of the functors $\mathrm{N}\colon \mathbf{sGrp} \rightarrow \mathbf{s^2Set}$ and $\mathrm{Tot}\colon \mathbf{s^2Set} \rightarrow \mathbf{sSet}$, where $\mathbf{s^2Set}$ denotes the category of bisimplicial sets, where $\mathrm{N}$ denotes the nerve functor for simplicial groups (that is, nerve of groups, taken dimensionwise), and where $\mathrm{Tot}$ denotes the Artin-Mazur total simplicial set functor. On the other hand, one has the composite $\mathrm{Diag} \circ \mathrm{N}\colon \mathbf{sGrp} \rightarrow \mathbf{sSet}$, where $\mathrm{Diag}\colon \mathbf{s^2Set} \rightarrow \mathbf{sSet}$ denotes the diagonal simplicial set functor. Already $\mathrm{Tot} X$ and $\mathrm{Diag} X$ for a bisimplicial set $X$ are (naturally) weakly homotopy equivalent simplicial sets, see [1], so in particular $\overline{\mathrm{W}} G \cong \mathrm{Tot} \mathrm{N} G$ and $\mathrm{Diag} \mathrm{N} G$ for a simplicial group $G$ are weakly homotopy equivalent simplicial sets. In fact, $\overline{\mathrm{W}} G$ and $\mathrm{Diag} \mathrm{N} G$ for a simplicial group $G$ are (simplicially) homotopy equivalent simplicial sets, see [3]. For the isomorphism $\overline{\mathrm{W}} G \cong \mathrm{Tot} \mathrm{N} G$, see e.g. [2, rem. 4.19].

This could help for your question if your nerve functor $\mathbf{Crs} \rightarrow \mathbf{sSet}$ is (up to isomorphy? or at least up to natural weak homotopy?) the composite $\mathrm{Diag} \circ \mathrm{N} \circ \mathrm{T}$.

[1] Cegarra, A.M.; Remedios, J.: The relationship between the diagonal and the bar constructions on a bisimplicial set, Topology and its Applications 153(1) (2005), pp. 21-51. doi:10.1016/j.topol.2004.12.003

[2] Thomas, S.: (Co)homology of crossed modules, Diploma Thesis, RWTH Aachen, 2007. http://www.math.rwth-aachen.de/~Sebastian.Thomas/publications/

[3] Thomas, S.: The functors Wbar and Diag Nerve are simplicially homotopy equivalent, Journal of Homotopy and Related Structures 3(1) (2008), pp. 359-378. http://www.math.rwth-aachen.de/~Sebastian.Thomas/publications/

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Just to check: are you defining the nerve of a group to be the nerve of the associated 1-object groupoid? –  David Roberts Sep 20 '10 at 4:24
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Yes. (Perhaps I use a non-standard convention numbering the morphisms in reverse order; that was to get the formulas of the simplicial operations $\overline{W} G$ as originally defined by Kan.) –  Sebastian Thomas Sep 20 '10 at 16:53

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