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](https://ncatlab.org/nlab/show/simplicial+localization#hammock_localization), 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?



Is the hom-simplicial set in the hammock localization a nerve?
The nLab description is not correct. 


For each "shape" of zig-zag, there is a "hammock category" for it (not a groupoid, and the nLab page I am looking at never mentions groupoids here), whose objects are functors $f\colon Z\to C$ ($Z$ is an abstract zig-zag of a particular shape) such that the backwards arrows of $Z$ are sent into $W$. The morphisms are natural transformations $f\to f'$ which are identities at the ends (and which in the original formulation of Dwyer and Kan are such that the vertical arrows of the transformation must also be in $W$, though this condition turns out not to really be necessary, so it is nowadays often dropped). 


The full hammock mapping space $L^H(X,Y)$ is a quotient of all the nerves of these hammock categories by an equivalence relation which is *not* compatible with the category structure (though it is compatible with the simplicial structure). Thus, $L^H(X,Y)$ is not a nerve of a category.


Go look at the original Dwyer-Kan paper, or at the book by Dwyer-Hirschhorn-Kan-Smith.


