User sebastian thomas - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T01:53:41Z http://mathoverflow.net/feeds/user/9303 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/44047/localizing-an-arbitrary-additive-category/44104#44104 Answer by Sebastian Thomas for Localizing an arbitrary additive category Sebastian Thomas 2010-10-29T10:58:12Z 2010-10-29T10:58:12Z <p>In a particular context, I have given a criterion in section 6 of "On the 3-arrow calculus for homotopy categories" (available at <a href="http://www.math.rwth-aachen.de/~Sebastian.Thomas/publications/" rel="nofollow">http://www.math.rwth-aachen.de/~Sebastian.Thomas/publications/</a>). In this context, every morphism is represented by a 3-arrow, that is, a formal inverse followed by an morphism in the original category followed by a formal inverse.</p> <p>Could you give more details of your example?</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> http://mathoverflow.net/questions/34193/nerves-of-crossed-complexes-group-t-complexes-and-classifying-spaces/39105#39105 Comment by Sebastian Thomas Sebastian Thomas 2010-09-20T16:53:57Z 2010-09-20T16:53:57Z 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.)