Timeline for Is the hom-simplicial set in the hammock localization a nerve?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 16, 2016 at 14:10 | vote | accept | Bruno Stonek | ||
| Feb 21, 2016 at 10:40 | comment | added | Mike Shulman | @CharlesRezk Thanks, that's what I wanted! | |
| Feb 21, 2016 at 2:39 | comment | added | Charles Rezk | ... 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 2:38 | comment | added | Charles Rezk | @MikeShulman Don't know how to build an example. But the problem is this: consider two factorizations $f_1\cdots f_r=g_1\cdots g_s$ of the same map in $C$. In $(L^H)_0(X,Y)$, the map can be represented by a zig-zag of length one. But it is also equivalent to zig-zags of length $2r+1$ and $2s+1$, by inserting "backwards identities" into the two factorizations. ..... | |
| Feb 20, 2016 at 23:30 | comment | added | Mike Shulman | I would be curious to see an example showing why the equivalence relation in question is not compatible with the category structure; do you have one in mind? | |
| Feb 20, 2016 at 22:50 | comment | added | Zhen Lin | 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 22:31 | comment | added | Charles Rezk | @ZhenLin I didn't remember that (and they don't do it in the draft copy I have on hand). | |
| Feb 20, 2016 at 21:41 | comment | added | Zhen Lin | In fact, in DHKS, they construct precisely such a 2-category as a model for simplicial localisation. | |
| Feb 20, 2016 at 19:56 | comment | added | Charles Rezk | @YonatanHarpaz You can do that; I don't know if that nerve model would be weakly equivalent to $L^H(X,Y)$. The other question is: would this nerve model lead to a simplicial category like $L^H$, i.e., does the composition law $L^H(X,Y) \times L^H(Y,Z)\to L^H(X,Z)$ still make sense for your variant? | |
| Feb 20, 2016 at 19:02 | comment | added | Yonatan Harpaz | 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 17:08 | comment | added | Todd Trimble | 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 16:51 | history | answered | Charles Rezk | CC BY-SA 3.0 |