Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorronniegpd
   • CommentTimeAug 3rd 2015
   • PermaLink
   Author: ronniegpd
   Format: TextI am not too happy with the current page on Higher van Kampen theorems as Lurie's book confuses the van Kampen theorem with the small simplex theorem, on which there is a nice page, and there is no evidence that the Higher Homotopy van Kampen theorems on which I have published can be deduced from results in Lurie's books. I have a recent exposition on "A philosophy of cmodelling and computing homotopy types" (given at CT2015, Aveiro) which is available on my preprint page and the CT2015 page http://pages.bangor.ac.uk/~mas010/brownpr.html whose abstract is as follows: ------------------------------------ This philosophy involves homotopically defined functors H from (Topological Data) to (Algebraic Data), and conversely "classifying space" functors B from (Algebraic Data) to (Topological Data). These should satisfy: 1. H is homotopically defined. 2. HB is naturally equivalent to 1. 3. The Topological Data has a notion of connected. 4. For all Algebraic Data A, we have BA is connected. 5. H preserves certain colimits of connected Topological Data. The algebraic data splits into several equivalent kinds, ranging from "broad" to "narrow", related by Dold-Kan type equivalences. The broad data is used for conjecturing and proving theorems; the narrow data is used for calculations and relating to classical methods. As examples of Algebraic Data we give groupoids, crossed modules and crossed squares. We give a sample computation, using crossed squares, of the homotopy 3-type of the mapping cone of the classifying space of a morphism of crossed modules. --------------------------------- So I am thinking of a new page on this. Any comments? Ronnie

   I am not too happy with the current page on Higher van Kampen theorems as Lurie's book confuses the van Kampen theorem with the small simplex theorem, on which there is a nice page,
   and there is no evidence that the Higher Homotopy van Kampen theorems on which I have published can be deduced from results in Lurie's books.
   
   I have a recent exposition on "A philosophy of cmodelling and computing homotopy types" (given at CT2015, Aveiro) which is available on my preprint page and the CT2015 page
   http://pages.bangor.ac.uk/~mas010/brownpr.html
   whose abstract is as follows:
   ------------------------------------
   This philosophy involves homotopically defined functors H from (Topological Data) to (Algebraic Data), and conversely "classifying space" functors B from (Algebraic Data) to (Topological Data). These should satisfy:
   1. H is homotopically defined.
   2. HB is naturally equivalent to 1.
   3. The Topological Data has a notion of connected.
   4. For all Algebraic Data A, we have BA is connected.
   5. H preserves certain colimits of connected Topological Data.
   
   The algebraic data splits into several equivalent kinds, ranging from "broad" to "narrow", related by Dold-Kan type equivalences. The broad data is used for conjecturing and proving theorems; the narrow data is used for calculations and relating to classical methods.
   
   As examples of Algebraic Data we give groupoids, crossed modules and crossed squares. We give a sample computation, using crossed squares, of the homotopy 3-type of the mapping cone of the classifying space of a morphism of crossed modules.
   ---------------------------------
   
   So I am thinking of a new page on this.
   
   Any comments?
   
   Ronnie
2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeAug 4th 2015
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexAre you talking about the page [[higher homotopy van Kampen theorem]] or [[higher van Kampen theorem]]? (Why are those separate pages anyway?) In any case, I would suggest rather than "forking" a new page, to instead add material to the existing page. You have a particular opinion about what "higher van Kampen theorems" should mean, but I think there are other equally valid opinions.

   Are you talking about the page higher homotopy van Kampen theorem or higher van Kampen theorem? (Why are those separate pages anyway?)

   In any case, I would suggest rather than “forking” a new page, to instead add material to the existing page. You have a particular opinion about what “higher van Kampen theorems” should mean, but I think there are other equally valid opinions.