Timeline for Is the singular simplicial functor full
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8 events
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| Feb 21, 2023 at 14:17 | comment | added | user494312 | Is there a positive result that under some assumptions $\text{Sing} X$ is full? e.g. that for CW complexes $X$ and $Y$ $Hom_{Top}(X,Y)=Hom_{sSets}(Sing X, Sing Y)$? | |
| May 20, 2019 at 9:43 | comment | added | Denis Nardin | @StefanoNicotra It might be false, you're right that looking at it today it seems a lot fishier to me (maybe I was thinking of homotopical Kan extensions when I wrote that comment...). In particular the counit is almost certainly $|\mathrm{Sing}X|→X$... | |
| May 20, 2019 at 9:40 | comment | added | Stefano Nicotra | @DenisNardin Is that true? I thought that every numerically generated space was a canonical colimit over ∆, where we consider all possible morphisms between standard simplices and not just the realisations of the ones coming from ∆. Besides, unless I'm missing something, the Kan extension described in the Question should be just |Sing(X)|, which is a CW-complex so any topological space which is not a CW-complex should be a counterexample (and even if you start with a CW-complex you usually end up with a much bigger one after applying |Sing(-)|). | |
| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Mar 27, 2017 at 11:32 | answer | added | Peter LeFanu Lumsdaine | timeline score: 6 | |
| Feb 11, 2017 at 8:40 | history | edited | user24453 | CC BY-SA 3.0 |
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| Feb 11, 2017 at 4:04 | comment | added | Denis Nardin | I suspect that the left Kan extension is the identity for numerically generated spaces (which are the kind of spaces you want to consider in homotopy theory anyway). | |
| Feb 11, 2017 at 3:50 | history | asked | user24453 | CC BY-SA 3.0 |