nerves of crossed complexes, group T-complexes and classifying spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T13:23:41Z http://mathoverflow.net/feeds/question/34193 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/34193/nerves-of-crossed-complexes-group-t-complexes-and-classifying-spaces nerves of crossed complexes, group T-complexes and classifying spaces David Roberts 2010-08-02T01:44:44Z 2010-09-17T14:04:32Z <p>A (reduced) <a href="http://ncatlab.org/nlab/show/crossed+complex" rel="nofollow">crossed complex</a> 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.</p> <p>There are a couple of constructions that could be called a classifying space of a crossed complex, namely the one given at <a href="http://mathoverflow.net/questions/2734/classifying-space-of-a-crossed-complex" rel="nofollow">this MO question</a>, and the simplicial set arising from the $\bar{W}$ functor applied to the simplicial group which is the <a href="http://ncatlab.org/nlab/show/group+T-complex" rel="nofollow">group T-complex</a> associated to the crossed complex. (Aside: I would be tempted to call this the <em>groupal nerve</em>, 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.</p> <p>My question is this:</p> <p>Does the usual classifying space functor from crossed complexes to topological spaces lift, up to homotopy, through the functor $|\bar{W} - |: sGrp \to Top$?</p> <hr> <p>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 <a href="http://mathoverflow.net/questions/2734/classifying-space-of-a-crossed-complex" rel="nofollow">nerve</a> functor and $\bar{W}:sGrp \to sSet$ the classifying space functor, do we have a (weak) homotopy equivalence $$\bar{W}T(G) \sim NG?$$</p> http://mathoverflow.net/questions/34193/nerves-of-crossed-complexes-group-t-complexes-and-classifying-spaces/39105#39105 Answer by Sebastian Thomas for nerves of crossed complexes, group T-complexes and classifying spaces Sebastian Thomas 2010-09-17T14:04:32Z 2010-09-17T14:04:32Z <p>Perhaps the following is related to your question:</p> <p>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].</p> <p>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 $\mathrm{Diag} \circ \mathrm{N} \circ \mathrm{T}$.</p> <p>[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</p> <p>[2] Thomas, S.: (Co)homology of crossed modules, Diploma Thesis, RWTH Aachen, 2007. <a href="http://www.math.rwth-aachen.de/~Sebastian.Thomas/publications/" rel="nofollow">http://www.math.rwth-aachen.de/~Sebastian.Thomas/publications/</a></p> <p>[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. <a href="http://www.math.rwth-aachen.de/~Sebastian.Thomas/publications/" rel="nofollow">http://www.math.rwth-aachen.de/~Sebastian.Thomas/publications/</a></p>