Let $(C,w)$ be a relative category. Then associated to it we have its hammock localization, $L^H(C,w)$, which is a simplicially enriched category.

If $X,Y\in C$, the description of the simplicial set $L^H(C,w)(X,Y)$ can be found in Dwyer-Kan, "Calculating Simplicial Localizations", 2.1. The 0-simplices are zig-zags where the reversed arrows are in $w$. The $k$-simplices are defined by taking "natural transformations" of such zig-zags that fix the endpoints, all go in the same direction, and are in $w$. (See the article for more details).

In the nLab, it is claimed that this is the nerve of a certain category (groupoid, actually), whose objects are some equivalence classes of zig-zags (under an equivalence relation whose formulation I don't really understand), and whose morphisms are similar to the 1-simplices of the simplicial set above.

Is this formulation correct? I'm failing to see whether they're equal (or equivalent). In particular, I'm troubled by the fact that the $\pi_0$ of the nerve of a groupoid gives you the set of *isomorphism classes* of the groupoid, so I'm nervous about whether $\pi_0$ of the nerve defined in the nLab will actually really give $C[w^{-1}](X,Y)$.

If the formulation is not correct, the next question would be: is the simplicial set $L^H(C,w)(X,Y)$ defined by Dwyer-Kan the nerve of some category?

notthe nerve of a groupoid. If it were then $(\infty, 1)$-categories would be the same as $(2, 1)$-categories. $\endgroup$