Modern proofs for simplicial localizations

Question

I know that the references usually regarded as standard for simplicial localizations are the Dwyer and Kan's three articles from the 80's. I would be interested in a more modern approach to the subject, and in particular the following two issues keep surfacing in my work, for which I'd appreciate to be able to cite something that is more compact and student-friendly than the original works.

1. Define the $\infty$-category $L(\mathcal{C}, \mathcal{W})$ associated to a relative category $(\mathcal{C}, \mathcal{W})$ as the homotopy pushout of the span $\coprod_{\mathcal{W}} J \leftarrow \coprod_{\mathcal{W}} \Delta^1 \rightarrow N(\mathcal{C})$ in $\textbf{sSet}$ (or anything equivalent to it). I would like to prove that for a simplicial model category $\underline{\mathbf{M}}$ then there is an equivalence of $\infty$-categories $$ \text{N}_{\Delta}(\underline{\mathbf{M}}^{cf}) \simeq L(\mathbf{M}, \mathcal{W}) $$ where $\mathbf{M}$ is the underlying (unenriched) category of $\underline{\mathbf{M}}$. Ideally, I would like to avoid the three above mentioned articles, preferring instead later developed and thoroughly studied technologies such as the equivalence between Joyal and Bergner model structures on $\mathbf{sSet}$ and $\mathbf{sCat}$, or that between Barwick-Kan and Rezk model structures on $\mathbf{RelCat}$ and $\mathbf{ssSet}$, or the like.

2. I suspect that, for an arbitrary relative category, there should be an equivalence $$\text{N}_{\Delta}(L^H(\mathcal{C}, \mathcal{W})) \simeq L(\mathcal{C}, \mathcal{W})$$ where $L^H$ denotes the hammock localization, and $L$ is the associated $\infty$-category as defined above, but I can't find a proof of this. Again, a modern approach using high technology is preferable to the original paper where the hammock localization was introduced.

Answer 1

For Question 1. It is documented in Corolary 4.2.4.8 in Lurie's Higher Topos theory that $N_\Delta(\underline{M}^{cf})$ is an $\infty$-category with small limits (and small colimits). Moreover, the inclusion $M^{cf}\subset \underline{M}^{cf}$ induces a functor $$(*)\qquad L(M,W)\cong L(M^{cf},W\cap M^{cf})\to N_\Delta(\underline{M}^{cf})\, .$$ The homotopy $1$-category of the $\infty$-category $N_\Delta(\underline{M}^{cf})$ is the ordinary $1$-localization of $M^{cf}$ by weak equivalences. This means that the comparison functor above induces an equivalence on homotopy categories. On the other hand, since, by virtue of Proposition 7.7.4 of Higher categories and homotopical algebra, $L(M,W)$ also has small (co)limits, it is sufficient to prove that the functor $(*)$ above commutes with finite limits, by Theorem 6.7.10 from loc. cit. The fact that the terminal object is preserved is obvious. Using Proposition 7.5.6 (as well as Theorem 7.5.18) from loc. cit., this amounts to check that pullback squares along fibrations between fibrant objects do provide pulback squares in $N_\Delta(\underline{M}^{cf})$, which follows from Theorem 4.2.4.1 in Higher Topos theory.

In fact, with the arguments above, one may characterize $L(M,W)$ as follows: a finitely complete $\infty$-category $M_\infty$ equipped with a functor $\gamma: N(M)\to M_\infty$ is the localization of $M$ by $W$ if and only if the following conditions are verified:

1. $\gamma$ sends any element of $W$ to an invertible map in $M_\infty$;
2. the induced functor $M\to ho(M_\infty)$ is the $1$-localization of $M$ by weak equivalences;
3. the functor preserves terminal objects and sends any pullback square of fibrant objects along fibrations in $M$ to pullback squares in $M_\infty$.

(the latter characterization can be extended to all kind of variations on model structures that I know of: semi-model structures, categories of fibrant objects, and so on).

For Question 2. I insist that checking that the hammock localization of Dwyer and Kan has the universal property of $L(C,W)$ really is obvious from the original construction via the Quillen equivalence relating relative categories, simplicial categories, and quasi-categories, just by reading the original sources. But there are documented more recent references such as Hinich's Dwyer-Kan localization revisited, for instance.