Timeline for Does the category of strict $2$-categories together with Dwyer-Kan equivalences provide a model for $(\infty,1)$-categories?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 31, 2022 at 3:24 | answer | added | Dmitri Pavlov | timeline score: 1 | |
| Dec 26, 2011 at 8:37 | comment | added | Jonathan Chiche | OK, thanks a lot! I will try to write up all the details and make the proof available as soon as possible for those interested. | |
| Dec 26, 2011 at 1:17 | comment | added | Clark Barwick | I don't really mean to suggest that you use such a model structure on $2-\mathrm{Cat}$. The functor given by applying the nerve to each $\mathrm{Hom}$-category is already a weak equivalence-preserving functor $2-\mathrm{Cat}\to\mathrm{S}-\mathrm{Cat}$, and I claim that this is an equivalence of homotopy theories. To obtain a homotopy inverse, choose a homotopy inverse $\Gamma$ to the nerve [in the sense of the brilliant paper of Fritsch and Latch (MR0612870)] that is lax monoidal in which $\Gamma(X)\times\Gamma(Y)\to\Gamma(X\times Y)$ is an equivalence, and apply it to the mapping spaces. | |
| Dec 25, 2011 at 18:23 | comment | added | Jonathan Chiche | Thanks! I am aware of the Quillen equivalence which you mention, but I fail to see right now the details as to how it lifts to an equivalence between $2-Cat$ and the category of simplicial categories. When I saw your comment I first thought there was some result of which I was not aware, stating that a Quillen equivalence could be lifted up to the level of enriched categories under some assumptions, but as far as I know there is no known model structure on $2-Cat$ whose weak equivalences are Dwyer-Kan equivalences, so I am probably really missing something, and I am afraid it may be obvious. | |
| Dec 25, 2011 at 16:20 | comment | added | Clark Barwick | I think it's actually easier than what Kan and I do. There is a right Quillen equivalence $\mathrm{Cat}\to\mathrm{S}$ --- where $\mathrm{Cat}$ is equipped with its Thomason model structure and $\mathrm{S}$ is the usual relative category of simplicial sets ---, which is given by $\mathrm{Ex}^2$ of the nerve. This induces a functor $2-\mathrm{Cat}\to \mathrm{S}-\mathrm{Cat}$, which can be seen to be an equivalence of homotopy theories. | |
| Dec 25, 2011 at 15:56 | comment | added | Chris Schommer-Pries | Interesting question! I missread the question the first time around, so have deleted my non-answer. The content was just that the papers of Barwick and Kan (which can easily be found on the ArXiv) are related to this. | |
| Dec 25, 2011 at 9:32 | history | edited | Jonathan Chiche | CC BY-SA 3.0 |
Tried to add underscores once again...
|
| Dec 25, 2011 at 9:29 | comment | added | Jonathan Chiche | Ah, thank you, Angelo! Looks like underscores were suppressed. (I am sure they were in the input.) If this phenomenon is explained somewhere, I would be grateful if someone could provide a link so as to avoid that next time. | |
| Dec 25, 2011 at 9:26 | comment | added | Jonathan Chiche | I do not understand what is wrong with the LaTeX input. (There is no problem when I compile it.) Please someone tell me or edit the source accordingly, thanks! | |
| Dec 25, 2011 at 9:21 | history | edited | Angelo | CC BY-SA 3.0 |
added 3 characters in body
|
| Dec 25, 2011 at 9:17 | history | asked | Jonathan Chiche | CC BY-SA 3.0 |