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*