Timeline for Is the inclusion of its 2-skeleton into the walking idempotent homotopy cofinal?
Current License: CC BY-SA 4.0
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| Mar 20, 2020 at 1:06 | vote | accept | Tim Campion | ||
| Feb 29, 2020 at 10:08 | comment | added | Yonatan Harpaz | Another proof of this fact using Quillen cohomology of categories and obstruction theory appears in Example 3.3.11 of londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/topo.12074 (there is a free version on my site, but I see that the arxiv version is not up to date though). Roughly speaking, the point is that $Idem$ has low "Quillen cohomological dimension", and so once you know the first few coherences, the rest of the obstructions are zero. | |
| Feb 26, 2020 at 22:49 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Feb 26, 2020 at 22:46 | answer | added | Tim Campion | timeline score: 2 | |
| Feb 26, 2020 at 20:35 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Feb 26, 2020 at 20:12 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Feb 26, 2020 at 20:05 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Feb 26, 2020 at 18:59 | comment | added | Tim Campion | Something is wrong -- a fundamental group / homology computation shows that the weak homotopy type of the $n$-skeleton of $Idem$ is $S^n$ for $n$ odd, and weakly contractible for $n$ even. $Idem$ itself is weakly contractible, and homotopy cofinal functors preserve weak homotopy type, so the inclusion of the $n$-skeleton can only be homotopy cofinal for $n$ even. (This condition is necessary but not sufficient -- e.g. the 0 and 2 -skeleton inclusions are certainly not homotopy cofinal). In particular, the 3-skeleton is not homotopy cofinal in $Idem$, but perhaps the 4-skeleton is. | |
| Feb 26, 2020 at 18:22 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Feb 26, 2020 at 18:04 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Feb 26, 2020 at 17:58 | history | asked | Tim Campion | CC BY-SA 4.0 |