Timeline for Geometric models for classifying spaces of a group
Current License: CC BY-SA 2.5
19 events
| when toggle format | what | by | license | comment | |
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| S Jun 20, 2016 at 19:46 | history | suggested | Ali Taghavi |
I add a tag
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| Jun 20, 2016 at 19:35 | review | Suggested edits | |||
| S Jun 20, 2016 at 19:46 | |||||
| Aug 12, 2015 at 11:45 | answer | added | Tom Bachmann | timeline score: 1 | |
| Oct 29, 2010 at 3:55 | comment | added | Harry Gindi | @SB: Dugger and Spivak note in their paper on rigidification of mapping spaces that Lurie's nasty construction of \mathfrak{C} on simplices is simply the bar construction as I wrote up in my answer. I suspect that Lurie was trying to avoid the language of monads in chapter 1, which is written for a "general audience" | |
| Oct 29, 2010 at 3:25 | answer | added | Harry Gindi | timeline score: 4 | |
| Oct 29, 2010 at 3:22 | comment | added | David Roberts♦ | @SB Thanks for clearing that up! I didn't notice our comments cross paths until now. | |
| Oct 29, 2010 at 3:11 | history | edited | Somnath Basu | CC BY-SA 2.5 |
added 190 characters in body
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| Oct 29, 2010 at 3:08 | comment | added | David Roberts♦ | @Harry, I know, but I'm thinking it's not what's described in the question, and I'm [feeling lazy/not in a position to check at the moment]. | |
| Oct 29, 2010 at 3:08 | comment | added | Somnath Basu | @Harry - The description given in Lurie is the one I intended. Although, right now I'm trying to parse Lurie's definition and see what I get for known examples. | |
| Oct 29, 2010 at 2:53 | comment | added | Harry Gindi | @DR: The construction I described is just restricting the "associated quasicategory" functor for topological categories to topological groups. | |
| Oct 29, 2010 at 2:45 | comment | added | David Roberts♦ | @Harry - I originally thought that $\mathcal{N}$ as described was related to $S(-):Top \to sSet$ applied to the arrows of the Top-groupoid $\mathbf{B}G$, but then I wasn't sure. Does your description arrive at the 2-simplices as in the question? | |
| Oct 29, 2010 at 1:47 | comment | added | Harry Gindi | Also, if you've read HTT, $FU_\cdot$ is called $\mathfrak{C}$ by Lurie. | |
| Oct 29, 2010 at 1:45 | comment | added | Harry Gindi | By homotopy-coherent nerve, define $\mathcal{N}(C)_n:=Hom_{Cat_\Delta}(FU_\cdot([n]),C)$ where $FU_\cdot$ is the comonad resolution (that is, the bar construction for the free-category comonad). | |
| Oct 29, 2010 at 1:41 | comment | added | Harry Gindi | What's wrong with doing the following: Consider a topological group to be a Top-enriched category with one object, apply the total singular complex functor to each homspace, then take the homotopy-coherent nerve of this sSet-enriched category. | |
| Oct 29, 2010 at 1:18 | comment | added | David Carchedi | Maybe I'm confused, what IS the definition of topological nerve, if its not just the enriched nerve? That's what I meant. The only definition of "topological nerve" I know is giving each set $N(C)_n$ coming from the nerve of the underlying category with the subspace topology inherited from $C_1^n$. So, if not this, what do you mean? | |
| Oct 29, 2010 at 1:13 | comment | added | David Roberts♦ | They're not the same. The 'topological nerve' looks like a homotopy version of the usual nerve, and contains it as a sub-simplicial space. For example, the space of 2-simplices for NG is G\times G, but the space of 2-simplices for \cal{N}G is (G x G) x_G G^I. But to be concrete, how are the 3-simplices formed? There are several choices I can think of. Also, how is the 'topological nerve' formed for general categories in Top? It's not a straightforward choice (Top-enriched categories are straightforward, though) | |
| Oct 29, 2010 at 1:03 | answer | added | David Carchedi | timeline score: 2 | |
| Oct 29, 2010 at 0:52 | comment | added | David Carchedi | How is your "usual nerve" any different all from your topological nerve? | |
| Oct 29, 2010 at 0:12 | history | asked | Somnath Basu | CC BY-SA 2.5 |