Timeline for Waldhausen $K$-theory before group completion
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 26, 2018 at 20:34 | vote | accept | Tim Campion | ||
| Jun 24, 2018 at 18:17 | answer | added | Yonatan Harpaz | timeline score: 6 | |
| Jun 13, 2017 at 4:20 | comment | added | Tim Campion | Hi Elden! Yeah, I just mean "localize at the weak equivalences" (in the $\infty$-categorical sense). So this is the most vanilla of the above functors, the only twist is remembering the symmetric monoidal structure. If you model $\infty$-categories by simplicially-enriched categories, it's Dwyer-Kan localization; if you model $\infty$-categories by marked simplicial sets, it's fibrant replacement of (nerve of category, weak equivalences), etc. In particular, it forgets the cofibration part of the Waldhausen structure (though cofibrations still figure in the definition of $\mathcal{E} W$). | |
| Jun 13, 2017 at 3:27 | comment | added | Elden Elmanto | Hi Tim. I don't quite understand the functor in bullet point 1. Let's say we consider the Waldausen category $Perf_k$. Maybe we regard this as a discrete Waldhausen category (though you can also do $\infty$-Waldhausen). What do you send this to? Something like invert all the weak equivalences or something? | |
| Jan 3, 2017 at 18:30 | comment | added | Tim Campion | @TylerLawson When I say "$K$ is a reflection of $K_\oplus$", I mean there is a map $K_\oplus \to K$ with the obvious universal property. I see that Waldhausen does prove that $S_\bullet$ is additive, but I don't see him proving a universal property. I see your point about $E_\infty$-valued "loops" -- I suppose one way to do it would be to fibrantly replace $wS_\bullet \mathcal{C}$ with a 1-object Segal space (regarding the "$\bullet$" direction as the "categorical" direction) and then take the endomorphism object, although getting the full $E_\infty$ structure will still take more work. | |
| Jan 3, 2017 at 16:45 | comment | added | Tyler Lawson | I believe that the answer to 2 is yes and that this is in Waldhausen's foundational paper (usually phrased as saying that one iterate of the S. construction suffices). However the S. construction implicitly deloops once. If you want to recover K-theory to have to take loops on S., making it difficult to get something that's not group-complete. | |
| Jan 3, 2017 at 6:05 | history | asked | Tim Campion | CC BY-SA 3.0 |