Timeline for What is the intuitive difference between these two simplicial subdivision functors?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Aug 7 at 15:25 | comment | added | John Rognes | @FShrike Yes, Fritsch and Puppe (1967) proved that for each simplicial set $X$ there exists a homeomorphism $|sd(X)| \cong |X|$, where sd is Kan's normal subdivision. See e.g. section 2.3 in mn.uio.no/math/personer/vit/rognes/papers/aoms186-nocrop.pdf for more history, references, and generalizations. We write Sd and B for your sd and S. | |
| Aug 7 at 15:11 | comment | added | FShrike | *to clarify: I know it is true for simplicial complexes, but don't know if it is true in total generality for all simplicial sets | |
| Aug 7 at 12:48 | comment | added | FShrike | @JohnRognes A lack of naturality would ruin the obvious argument for $|\mathrm{sd}(X)|\cong|X|$ (just unnaturally in $X$, on a case by case basis). Is this even true? I expect it should be true for the so-called simplicial complexes $X$, but am hesitant | |
| Oct 18, 2023 at 18:33 | vote | accept | FShrike | ||
| Oct 18, 2023 at 7:53 | comment | added | HenrikRüping | I posted the counterexample for naturality here: mathoverflow.net/a/139397/3969 | |
| Oct 18, 2023 at 0:22 | comment | added | FShrike | @JohnRognes Thank you for flagging that the canonical homeomorphism(s) isn't/aren't actually natural and in fact cannot be. That's what I get for checking naturality on face maps and not on degeneracy maps. I'll have a think about why that is. I'll also have a deeper think about the two answers given; thanks for sharing the paper you cowrote. | |
| Oct 18, 2023 at 0:21 | history | edited | FShrike | CC BY-SA 4.0 |
added 7 characters in body
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| Oct 17, 2023 at 21:12 | history | became hot network question | |||
| Oct 17, 2023 at 14:28 | answer | added | John Rognes | timeline score: 5 | |
| Oct 17, 2023 at 14:26 | comment | added | John Rognes | It is not true that |sd(X)| and |X| are naturally homeomorphic for simplicial sets X. No choices of homeomorphisms for X = \Delta^2 and X = \Delta^1 are simultaneously compatible with both degeneracy maps \Delta^2 \to \Delta^1. A weaker, quasi-naturality statement appears as Theorem 2.3.2 in mn.uio.no/math/personer/vit/rognes/papers/aoms186-nocrop.pdf . | |
| Oct 17, 2023 at 13:49 | answer | added | HenrikRüping | timeline score: 6 | |
| Oct 17, 2023 at 13:11 | history | asked | FShrike | CC BY-SA 4.0 |