Timeline for Is the singular simplicial complex functor $\operatorname{Sing}_\bullet:\operatorname{Top} \to \operatorname{sSets}$ fully faithful for nice spaces?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Mar 22, 2023 at 0:40 | vote | accept | user494312 | ||
| Mar 21, 2023 at 4:38 | comment | added | Reid Barton | What I meant is that, for fixed X, your equation holds for all Y if and only if the counit |Sing X| -> X is a homeomorphism. And both |Sing X| and (by assumption) X are "nice" spaces, so if (as usual) they are different, we should be able to see the difference using maps into another "nice" space Y. | |
| Mar 20, 2023 at 15:05 | comment | added | user494312 | Thank you. Though, is there a positive result saying that any counterexample is of this kind ? Say, comes from a natural transformation $|-|\implies|-|$ or something like that (actually, I do not think that your conterexample does arise in this way, so what I suggest is not enough.). In other words, to make formal your remark "Your question is implicitly about the counit $|SingX| \to X$." | |
| Mar 20, 2023 at 14:46 | comment | added | LSpice | The failure (though not the counterexample) is also mentioned in a comment of @ChrisSchommer-Pries. | |
| Mar 20, 2023 at 7:07 | history | answered | Reid Barton | CC BY-SA 4.0 |