Timeline for Multisimplicial geometric realization
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 18, 2013 at 5:29 | answer | added | Ricardo Andrade | timeline score: 2 | |
| Mar 18, 2013 at 3:16 | comment | added | Tom Goodwillie | Yes, Ricardo, that's what I meant to say. Dylan, the realization of a multisimplicial set is homeomorphic (not just homotopy equivalent) to the realization of its diagonal. | |
| Mar 18, 2013 at 2:46 | answer | added | Peter May | timeline score: 5 | |
| Mar 17, 2013 at 11:42 | comment | added | Dylan Wilson | @Tom: Is this different from what I said? (I'm asking because I'm actually curious if there's a difference, not because I think there's not.) | |
| Mar 17, 2013 at 9:20 | comment | added | Ricardo Andrade | I'm pretty sure Tom meant "the $(n,\ldots,n)$-multisimplices" instead of "the $(n_1,\ldots,n_k)$-multisimplices" in his comment. | |
| Mar 17, 2013 at 3:06 | comment | added | Tom Goodwillie | A $k$-fold multisimplicial set determines a simplicial set, its "diagonal", whose $n$-simplices are the $(n_1,\dots ,n_k)$-multisimplices. The realization of the latter is homeomorphic to that of the former. | |
| Mar 17, 2013 at 0:06 | comment | added | Dylan Wilson | Observations: $\Delta^{op}$ is a (homotopy) sifted category, so the diagonal map $\Delta^{op} \rightarrow \Delta^{op} \times \cdots \times \Delta^{op}$ is (homotopy) cofinal. It follows that computing homotopy colimits over the big guy is the same as computing homotopy colimits of the diagonal. So it's enough to show that the geometric realization of the diagonal is weakly equivalent to the space itself. But this simplicial set looks, at level $n$, like maps $\Delta^n \times \cdots \times \Delta^n \rightarrow X$. It's not obvious to me what to do from here... is the theorem even true? | |
| Mar 16, 2013 at 23:19 | history | asked | Jim McClure | CC BY-SA 3.0 |