Timeline for Modern proofs for simplicial localizations
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 22, 2021 at 11:52 | vote | accept | Giulio Lo Monaco | ||
| Apr 15, 2021 at 20:31 | answer | added | D.-C. Cisinski | timeline score: 12 | |
| Apr 15, 2021 at 15:35 | comment | added | Giulio Lo Monaco | @MikeShulman Thanks! This seems pretty much to answer my question 2. | |
| Apr 15, 2021 at 15:34 | comment | added | Giulio Lo Monaco | @Denis-CharlesCisinski Not really, in my opinion. In Lemma A.3.6.17, the category $\mathcal{C}$ is simplicially enriched and never regarded otherwise. The statement I want to reach is that we can forget about the simplicial structure, reobtain it via a nerve construction and get an equivalent simplicial category. In other words, I'm reasking mathoverflow.net/q/345044 with the additional assumption that $\mathcal{C}$ is tensored and cotensored over simplicial sets. | |
| Apr 14, 2021 at 15:05 | comment | added | Mike Shulman | Perhaps related: mathoverflow.net/q/92916/49 | |
| Apr 13, 2021 at 16:01 | comment | added | D.-C. Cisinski | I agree with Zhen Lin's comment above. I would not call the papers of Dywer and Kan "non-modern" and strongly encourage to read them. However, for 1., the papers of Dyyer and Kan are summarized (and generalized in Appendix A of Lurie's Higher Topos Theory. His definition of simplicial localization (see the begining of Section A.3.5) is automatically translated via the homotopy coherent nerve (or any Quillen equivalence relating simplicial categories and quasi-categories) as the definition of the localization you give above. Lemma A.3.6.17 essentially answers 1. | |
| Apr 13, 2021 at 13:15 | history | edited | David White | CC BY-SA 4.0 |
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| Apr 13, 2021 at 10:19 | comment | added | Zhen Lin | Personally I think the Dwyer–Kan papers were easier to understand (even if some crucial details were omitted...). Regarding your second question about arbitrary relative categories: this is basically already in the original papers. The point of the "standard" resolution of a category by free categories is that the groupoid completion of a free category is also the $\infty$-groupoid completion, so the "standard" localisation indeed presents the $\infty$-categorical localisation. Then they prove that the "standard" localisation is equivalent to the hammock localisation. | |
| Apr 13, 2021 at 9:37 | history | asked | Giulio Lo Monaco | CC BY-SA 4.0 |