# Is the hom-simplicial set in the hammock localization a nerve?

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?

• It is definitely not the nerve of a groupoid. If it were then $(\infty, 1)$-categories would be the same as $(2, 1)$-categories. Feb 19, 2016 at 21:13
• Are you sure the category described in the nLab is a groupoid? It doesn't sound like it to me (the vertical maps are in $w$, not isomorphisms). Feb 20, 2016 at 1:21
• @OmarAntolín-Camarena you're right, it's not a groupoid. Feb 20, 2016 at 7:06

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.

• I have mentioned this post in an nForum discussion: nforum.ncatlab.org/discussion/6972/… Hopefully the mistake will be repaired soon. Feb 20, 2016 at 17:08
• But couldn't these various zig-zag shapes themselves be organized into a category, such that if you have a morphism $p:Z \to Z'$ of zig-zag shapes then you have a restriction functor $p^*:Fun(Z',C) \to Fun(Z,C)$ in the other direction? In this case, if we call this category $Zig$, it seems that the nerve of the Grothendieck construction of the functor $F_{X,Y}:Zig^{op} \to Cat$ associated to a pair $X,Y \in C$ would be a natural small variant on the original definition that yields a nerve model for $L^H(X,Y)$. Feb 20, 2016 at 19:02
• In fact, in DHKS, they construct precisely such a 2-category as a model for simplicial localisation. Feb 20, 2016 at 21:41
• In §35 of the final version, they discuss a Grothendieck construction, use it to build a 2-category, and show that it is weakly equivalent to the hammock localisation. As far as I understand it, it boils down to Thomason's homotopy colimit theorem and the fact that hom-spaces of the hammock localisation are colimits of Reedy-cofibrant diagrams. Feb 20, 2016 at 22:50
• ... Now suppose you have two $1$-simplices in $(L^H)_1(X,Y)$: the first, $a$, given as a morphism of zig-zags of length $2r+1$, with target $(1,f_r,1,\dots,f_1,1)$, and another, $b$, given as a morphism of zig-zags of length $2s+1$, with source $(1,g_s,1,\dots,g_1,1)$. How do you represent $ba$ as an element of $(L^H)_2(X,Y)$? Feb 21, 2016 at 2:39