Let $(C,W)$ be a category equipped with a subcategory of weak equivalences. Its "classification diagram" or "bisimplicial nerve" $N(C,W)$ is a bisimplicial set, for which $N(C,W)\_n$ is the nerve of the category of functors $[n]\to C$ and natural weak equivalences between them. If we fibrantly replace $N(C,W)$ in the complete Segal space model structure, we obtain an $(\infty,1)$-category. Is this $(\infty,1)$-category equivalent to the $(\infty,1)$-categorical localization of $C$ at $W$ (e.g. the simplicial hammock localization, or a fibrant replacement of its "marked simplicial nerve")?


I believe that when $C$ is a model category, this follows from results in the various papers of Dwyer and Kan on simplicial localization. It could be that the same is true for the general case, but I haven't yet been able to see how.



Does the classification diagram localize a category with weak equivalences?
It seems to me that the answer is yes. Here is a sketchy argument.


Let us fix some notations. I will write $i(X)$ for the maximal Kan subcomplex of a quasi-category $X$ and $Hom(A,B)$ for the internal Hom of simplicial sets $A$ and $B$. If $B$ is a quasi-category, then so is $Hom(A,B)$ for any simplicial set $A$, and I will write
$$i(A,B)=i(Hom(A,B))$$
It is a fact that (at least up to simplicial homotopy) any complete Segal space is of the form $i(\Delta^\ast,X)$ for a quasi-category $X$, where $\Delta^\ast$ is the standard cosimplicial object of simplicial sets (given by the Yoneda embedding); furthermore, the canonical inclusion $X\to i(\Delta^\ast,X)$ is a weal equivalence (where $X$ is considered as a (constant) simplicial quasi-category). Therefore, it is sufficient to understand morphisms of bisimplicial sets $N(C,W)\to i(\Delta^\ast,X)$ in the homotopy category of the Rezk model structure. The point is that these are homotopy classes of maps of bisimplicial sets (being maps from a cofibrant object to a fibrant one), and that those morphisms of bisimplicial sets can be understood rather explicitely in terms of morphisms of marked simplicial sets.


If $u:N(C)\to i(\Delta^\ast,X)$ is a morphism of bisimplicial sets, it is completely
determined by the morphism of simplicial sets $u\_0:N(C)\to X=i(\Delta^\ast,X)\_0$. On the other hand, the map $u$ sends any arrow of $W$ to an invertible $1$-simplex of $X$ if and only if the maps
$$N(Hom([n],C))=Hom(\Delta^n,N(C))\to Hom(\Delta^n,X)$$
induced by $u\_0$ define maps
$$N(C,W)\_n\to i(\Delta^n,X)$$
which in turn define a morphism of bisimplicial sets
$$N(C,W)\to i(\Delta^\ast,X)$$
Moreover, any map $N(C,W)\to i(\Delta^\ast,X)$ is obtained in this way (I guess that interpreting all this in terms of marked simplicial sets would help to write down all this in a cleaner way). It is then easy to deduce from there that $N(C,W)$ has the universal property of the localization of $N(C)$ by $N(W)$ in the homotopy category of the Rezk model structure: to deal with homotopies, you just have to replace $C$ by $C\times I$, where $I$ denotes the contractible groupoid with two objects (with an adequate subcategory of weak equivalences).


