Timeline for The reflexive free-category comonad-resolution is a cofibrant replacement of the discrete simplicial category associated with an ordinary category in the Bergner model structure on the category of small simplicial categories?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 21, 2011 at 14:52 | comment | added | Harry Gindi | You just use the framework and it proves the existence in about one second. It will suffice to read chapter 1. If you're interested in building model structures on presheaf categories, it's an indispensable theory. | |
| Dec 19, 2011 at 1:16 | comment | added | Akhil Mathew | Really? How so? I haven't read Cisinski's thesis; do you have a reference to the relevant statement? | |
| Dec 18, 2011 at 21:55 | comment | added | Harry Gindi | It's also easy to construct it using Cisinski's Asterisque 308, fwiw. | |
| Dec 18, 2011 at 17:13 | comment | added | Akhil Mathew | (Elegant and short, at least if you grant the claim that if $K$ is a simplicial set and $\mathcal{C}$ an $\infty$-category, then the equivalences in $\mathrm{Fun}(K, \mathcal{C})$ are the "pointwise" ones, which follows more easily from the formalism of marked simplicial sets in HTT ch. 3.) Anyway, I'm not really sure yet whether Lurie's theorems can be proved in a shorter manner using this -- it seems that in any event there's some hard work to be done. | |
| Dec 18, 2011 at 17:09 | comment | added | Akhil Mathew | Incidentally, there is a very elegant and short proof of the Joyal model structure in "The theory of quasi-categories and its applications" (or in his unpublished manuscript "The theory of quasi-categories I"), which makes it transparent that the fibrant objects are the $\infty$-categories. However, there the categorical equivalences are defined in a different manner, and it seems that proving that $\mathfrak{C}$ preserves them is then nontrivial. | |
| Dec 18, 2011 at 17:04 | comment | added | Harry Gindi | I'll take it, but using HTT 2.2.0.1 is kind of cheating =p. | |
| Dec 18, 2011 at 17:04 | vote | accept | Harry Gindi | ||
| Dec 17, 2011 at 16:18 | history | answered | Akhil Mathew | CC BY-SA 3.0 |