Timeline for Does the classification diagram localize a category with weak equivalences?
Current License: CC BY-SA 3.0
8 events
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| Sep 15, 2015 at 13:32 | comment | added | Chris Schommer-Pries | The Barwick-Kan model structure on relative categories is lifted from bisimplicial sets using the functor $N_\xi$. So $N_\xi$ (and hence $N$) is automatically a relative functor (aka homotopical). It preserves all weak equivalences (not just between fibrant objects). In fact that is how the weak equivalences between relative categories are defined. | |
| Sep 15, 2015 at 3:41 | comment | added | Aaron Mazel-Gee | @ChrisSchommer-Pries I might be missing something, but I don't quite follow: we only know that $N_\xi$ (and hence $N$) has the correct behavior on fibrant relative categories. Or perhaps is $N$ itself known to itself be a relative functor? | |
| Apr 4, 2012 at 19:10 | comment | added | Chris Schommer-Pries | @Mike: "Does this approach extend to "relative quasicategories" as well?" I imagine it will, but it won't be out-of-the-box like your original question. I know that in one of their papers Barwick and Kan consider an analog of some of this structure for relative simplicial categories. | |
| Apr 4, 2012 at 17:50 | comment | added | Mike Shulman | Hmm, and in fact the statement I wanted is more or less explicit in a 2009 draft of Barwick-Kan that I had sitting around, but which doesn't seem to be available any more, called "Relative categories; another model for the homotopy theory of homotopy theories part II: the weak equivalences". | |
| Apr 4, 2012 at 17:33 | comment | added | Mike Shulman | Does this approach extend to "relative quasicategories" as well? | |
| Apr 4, 2012 at 17:33 | comment | added | Mike Shulman | Ah, excellent! I had thought of using Toen's result, but I didn't realize that Barwick-Kan had also proved that hammock localization was an equivalence of homotopy theories. Too bad I can only accept one answer. | |
| Apr 4, 2012 at 16:18 | history | edited | Chris Schommer-Pries | CC BY-SA 3.0 |
Added summary.
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| Apr 4, 2012 at 16:03 | history | answered | Chris Schommer-Pries | CC BY-SA 3.0 |