Timeline for Thomason fibrant replacement and nerve of a localization
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 10, 2019 at 15:46 | vote | accept | Martin Frankland | ||
| Apr 2, 2019 at 0:48 | comment | added | Martin Frankland | Excellent. Thank you. | |
| Apr 1, 2019 at 20:25 | comment | added | D.-C. Cisinski | See Prop. 7.1.12 in my book "Higher categories...". There is also Prop. 7.3.8 which provides examples, and Theorem 4.4.36 gives a rather concrete characterization of proper functors. | |
| Apr 1, 2019 at 15:44 | comment | added | Martin Frankland | Thank you Denis-Charles! Could you point me to a reference about these and related topics? | |
| Apr 1, 2019 at 10:47 | comment | added | D.-C. Cisinski | About Question 2: if a functor $f:C\to D$ is smooth or proper with weakly contractible fibres, then it exhibits $D$ as the $(\infty,1)$-localization of $C$ by the maps which are sent to identities in$D$. This is true if $C$ and $D$ are $(\infty,1)$--categories, hence also for nerves of $1$-categories. Examples of smooth (proper) maps are Cartesian (coCartesian) fibrations. A typical instance of this is when $C$ is the category of (semi-)simplices of the nerve of $D$. | |
| Mar 31, 2019 at 21:42 | answer | added | Tim Campion | timeline score: 7 | |
| Mar 31, 2019 at 18:23 | comment | added | Martin Frankland | (cont'd) Furthermore, the nerve of the $1$-groupoidification is a much more drastic procedure. I think the nerve of the $1$-groupoidification $N(C[C^{-1}])$ is the Postnikov $1$-truncation of the $\infty$-groupoidification of the nerve $(NC)[NC_1^{-1}] \simeq \mathrm{Ex}^{\infty} NC$. Sorting out these issues is the goal of my question. | |
| Mar 31, 2019 at 18:19 | comment | added | Martin Frankland | @Dmitri Pavlov: Thank you for your comment. My question is mostly about distinguishing between 1-localization and $\infty$-localization. You mention the $\infty$-localization of a $1$-category $C$. With respect to which maps? Does a category come with an intrinsic notion of maps to invert? One could try "all maps", whose $\infty$-localization is the $\infty$-groupoidification, i.e., Kan fibrant replacement. This is a more drastic procedure than the Thomason fibrant replacement. | |
| Mar 31, 2019 at 14:01 | comment | added | Dmitri Pavlov | The fibrant replacement in the Thomason model structure is precisely the homotopical localization, alias ∞-localization. Thus, Question 1 immediately reduces to Question 2. | |
| Mar 31, 2019 at 5:30 | history | asked | Martin Frankland | CC BY-SA 4.0 |