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1. • CommentRowNumber1.
   • CommentAuthorMatanP
   • CommentTimeJun 13th 2013
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   Author: MatanP
   Format: Textadded an entry for the Kan Thurston Theorem. http://ncatlab.org/nlab/show/Kan-Thurston+Theorem

   added an entry for the Kan Thurston Theorem.
   http://ncatlab.org/nlab/show/Kan-Thurston+Theorem
2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 13th 2013
   • (edited Jun 13th 2013)
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   Author: Urs
   Format: MarkdownItexHi Matan, thanks! I have added some hyperlinks. By the way, you can easily link here to nLab entries by just the same syntax as on the nLab itself. So if you just type here [[Kan-Thurston theorem]] then it will come out properly hyperlinked: [[Kan-Thurston theorem]]

   Hi Matan,

   thanks!

   I have added some hyperlinks.

   By the way, you can easily link here to nLab entries by just the same syntax as on the nLab itself. So if you just type here

   [[Kan-Thurston theorem]]
   

   then it will come out properly hyperlinked:

   Kan-Thurston theorem

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 14th 2013
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   Author: DavidRoberts
   Format: MarkdownItexMy goodness, that's an astounding theorem, especially the last point.

   My goodness, that’s an astounding theorem, especially the last point.

4. • CommentRowNumber4.
   • CommentAuthorMatanP
   • CommentTimeJun 15th 2013
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   Author: MatanP
   Format: TextAdded an explenation of how a homotopy type is determined by the pair of groups. I agree that the last point is astounding but one should keep in mind that although we can model a given homotopy type with a pair of groups there is no chance of modeling the mapping space between two such with a corresponding pair of group maps. In that sense the review on mathscinet is misleading: group theory can seriously embedded in the homotopy theory of spaces, but Kan-Thurston theorem does not provide a converse embedding.

   Added an explenation of how a homotopy type is determined by the pair of groups.
   I agree that the last point is astounding but one should keep in mind that although we can model a given homotopy type with a pair of groups there is no chance of modeling the mapping space between two such with a corresponding pair of group maps. In that sense the review on mathscinet is misleading: group theory can seriously embedded in the homotopy theory of spaces, but Kan-Thurston theorem does not provide a converse embedding.
5. • CommentRowNumber5.
   • CommentAuthorCharles Rezk
   • CommentTimeJun 15th 2013
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   Author: Charles Rezk
   Format: MarkdownItexKan-Thurston does give an embedding of the homotopy category of spaces into the category of *presheaves* on the category of groups (more precisely, the category whose morphisms are conjugacy classes of homomorphisms). A space $X$ is sent to the presheaf $G \mapsto [BG,X]$.

   Kan-Thurston does give an embedding of the homotopy category of spaces into the category of presheaves on the category of groups (more precisely, the category whose morphisms are conjugacy classes of homomorphisms). A space $X$ is sent to the presheaf $G \mapsto [BG,X]$.