11
$\begingroup$

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?

$\endgroup$
3
  • 4
    $\begingroup$ It is definitely not the nerve of a groupoid. If it were then $(\infty, 1)$-categories would be the same as $(2, 1)$-categories. $\endgroup$ – Zhen Lin Feb 19 '16 at 21:13
  • 2
    $\begingroup$ 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). $\endgroup$ – Omar Antolín-Camarena Feb 20 '16 at 1:21
  • $\begingroup$ @OmarAntolín-Camarena you're right, it's not a groupoid. $\endgroup$ – Bruno Stonek Feb 20 '16 at 7:06
10
$\begingroup$

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.

$\endgroup$
10
  • 1
    $\begingroup$ I have mentioned this post in an nForum discussion: nforum.ncatlab.org/discussion/6972/… Hopefully the mistake will be repaired soon. $\endgroup$ – Todd Trimble Feb 20 '16 at 17:08
  • 1
    $\begingroup$ 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 $L^H(X,Y)$. $\endgroup$ – Yonatan Harpaz Feb 20 '16 at 19:02
  • 1
    $\begingroup$ In fact, in DHKS, they construct precisely such a 2-category as a model for simplicial localisation. $\endgroup$ – Zhen Lin Feb 20 '16 at 21:41
  • 3
    $\begingroup$ 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. $\endgroup$ – Zhen Lin Feb 20 '16 at 22:50
  • 1
    $\begingroup$ ... 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 $(L^H)_2(X,Y)$? $\endgroup$ – Charles Rezk Feb 21 '16 at 2:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.