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In a series of papers ([1], [2] and [3]), Dwyer and Kan introduced the hammock localization [2] as an effective technique to compute the simplicial localization of a model category [1]. This is meant to encode the higher homotopy implicit in a model category; indeed, if one takes $\pi_0$ hom-wise the result is the homotopy category of the original model category.

They say their main application is to reconstruct the function complexes between fibrant cofibrant objects in a simplicial model category. And they managed. Cool! On the other hand, a feature-oriented description of such a localization is given in [3]. Unfortunately, at that time (I guess!) the Lurie technology was not available, so a universal-property localization was not so easy to state in such a context. But now we do have it, so I wonder:

Given a model category $C$ with weak equivalences $\mathcal{W}$, consider its hammock localization $L^H(C, \mathcal{W})$. Take its derived homotopy coherent nerve $\textbf{R}N_{coh}(L^H(C,\mathcal{W}))$ and you will get a quasicategory. On the other hand, you can take the ordinary nerve of $C$, and then invert $\mathcal{W}$ in the sense of Lurie ([6], 1.3.4.1), obtaining $N(C)[\mathcal{W}^{-1}]$. Do these two constructions agree?

Recall that the derived homotopy coherent nerve is obtained by first taking a fibrant replacement (a.k.a., locally Kan simplicial category) and then applying the homotopy coherent nerve (which will automatically be fibrant in $\textrm{sSets}$, i.e., a quasicategory).

The furthest I could get. There is a nice paper by Hinich in which some of these issues are discussed. In particular, if $C$ is a fibrant simplicial (say model) category with weak equivalences $\mathcal{W}$, there is a weak equivalence of marked simplicial sets (proposition 1.2.1): $$ (**) \ \ \ (N_{coh}C, \mathcal{W}) \to \textbf{R}N(L^H(C,\mathcal{W}))^{\natural} $$ Notice also that the right-hand side is fibrant in the category of marked simplicial sets (over the point): indeed, from HTT 3.1.4.1 one can easily deduce that fibrant here means beaing a quasicategory with equivalences as marked edges. Also, in remark 1.3.4.2. from HA, we see that $N_{coh}C[\mathcal{W}^{-1}]$ can be identified with a fibrant replacement of $(N_{coh}C,\mathcal{W})$ in marked simplicial sets. Since equation (**) gives an explicit fibrant replacement, we get the thesis.

Unluckily, proposition 1.2.1 from Hinich is available only in the case in which $C$ is a (fibrant) simplicial category, though it's not clear where the simplicial structure is used. In fact, hammock localization is designed (also) to provide a simplicial structure on an arbitrary model category when such an extra structure does not come for free.

Thanks for your help! It's not really my field and I find it hard sometimes to orient in the choose-your-model yoga.

Bibliography

  • [1] - Simplicial localization of categories, Dwyer and Kan
  • [2] - Calculating simplicial localization, Dwyer and Kan
  • [3] - Function Complexes in Homotopical Algebra, Dwyer and Kan
  • [4] - DK localization revisited, Vladimir Hinich
  • [5] - Higher Topos Theory, Jacob Lurie
  • [6] - Higher Algebra, Jacob Lurie
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    $\begingroup$ This is related : mathoverflow.net/questions/390076/… $\endgroup$ Commented Jun 16, 2021 at 11:24
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    $\begingroup$ If $C$ is a 1-category, isn't it automatically a fibrant simplicial category (since sets are fibrant when seen as discrete simplicial sets)? $\endgroup$ Commented Jun 16, 2021 at 12:41
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    $\begingroup$ Dear Denis, you are a wise man. I think you are right. Possibly related: tinyurl.com/3nvvrpfe Should I close the question, since it is almost answered in the text? Or should I include your answer as an edit, for future references, or maybe you should post it and I accept it? $\endgroup$ Commented Jun 16, 2021 at 15:57
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    $\begingroup$ @AndreaMarino: The question should remain open, since it provides valuable references, and is clearly research-level. Also, URL shorteners like tinyurl should not be used on MathOverflow. $\endgroup$ Commented Jun 16, 2021 at 17:00
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    $\begingroup$ Sorry for the shortener, I didn't want to spoiler the joke :) well then I'll leave it open! $\endgroup$ Commented Jun 16, 2021 at 21:00

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It's generally best not to leave questions without an answer, even if they are answered in the comments. MO best practice is to post a CW answer summarizing the answer from the comments. In this case, the answer is "yes: these two constructions agree" because exactly what the OP sketched works, because $C$ is a 1-category hence automatically a fibrant simplicial category (with the discrete simplicial structure).

Aside: I stumbled upon this question just now when answering a different hammock localization question: https://mathoverflow.net/a/467080/11540. My only qualm about the present question is that I wouldn't call it a "localization in the sense of Lurie" or say "Lurie technology" or "invert $W$ in the sense of Lurie" for two reasons. First, this technology existed before Lurie. However, I agree that Lurie's book is a wonderful place to learn it. In that book, he himself points out that just because results are not attributed to others does not mean one should assume Lurie was the first to prove them. Secondly, there are several ways to invert things in Lurie's book, including passage from an $\infty$-category to its homotopy category and also Bousfield localization. So, I think it's best to be clear precisely what localization is being used, and I'm glad the OP pointed to the specific place 1.3.4.1.

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