Timeline for Is there a category whose isomorphisms are precisely the simple homotopy equivalences?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 16, 2014 at 4:54 | vote | accept | John Pardon | ||
| Nov 16, 2014 at 4:46 | comment | added | Tyler Lawson | Well, but I think that what Denis had in mind was something more like localizing the category of finite complexes with respect to the elementary expansions and collapses, and so I'd still like to see more explicitly how more maps become isomorphisms than composites of these. | |
| Nov 16, 2014 at 4:44 | comment | added | Denis Nardin | Yeah, I've checked it and you're right. I still think that part of the proof can be salvaged, but it's sort of a moot point since all I would prove is that the homotopy category is given by quotienting by the homotopy equivalence relation... | |
| Nov 16, 2014 at 4:41 | comment | added | Qiaochu Yuan | @Denis: in your answer you claimed that simple homotopy equivalences are compositions of simple expansions and simple contractions, but the definition I'm familiar with is that a simple homotopy equivalence is a map which is homotopic to such a composition. Your argument doesn't work with this second definition. | |
| Nov 16, 2014 at 4:37 | comment | added | Denis Nardin | Well, I assumed that the map $C\to C[s^{-1}]$ sends only the simple equivalences to invertible morphism, but apparently that's false :$ | |
| Nov 16, 2014 at 4:36 | comment | added | Tyler Lawson | @DenisNardin I haven't yet been able figure out where the incompatibility between our statements is. | |
| Nov 16, 2014 at 4:35 | comment | added | Denis Nardin | This seems to imply that the localization of finite complexes at the simple equivalences is the homotopy category (hence making my answer rubbish). | |
| Nov 16, 2014 at 4:30 | history | answered | Tyler Lawson | CC BY-SA 3.0 |