Timeline for On the paradox that $n$-coskeletal simplicial sets model all homotopy types
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8 events
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| Sep 19, 2018 at 15:16 | comment | added | Tim Campion | We have canonical acyclic cofibrations $X \to Ex^\infty X$ and $X \to Sing |X|$, and my thought was that there was a canonical factorization $Ex^\infty X \to Sing |X|$ which was injective, and a homotopy equivalence by 2/3. Then because $Ex^\infty X$ is fibrant, this map would have a retract. But now I'm not so sure there is such an injective factoriztion. Of course, your argument seems to be actually correct! | |
| Sep 19, 2018 at 14:13 | comment | added | S. Douteau | Disregard my previous comment. I still don't think $\text{Ex}^{\infty}X$ is a retract of $\text{Sing}|X|$, but this is not needed to show the existence of a lift. Take any weak equivalence $\text{Ex}^{\infty}X\to \text{Sing}|X|$ and factor it through some space $Z$ as a trivial cofibration followed by a trivial fibration. Now $Z$ is a retract of $\text{Ex}^{\infty}X$ and $Z\to \text{Sing}|X|$ is a trivial fibration. This should provide the existence of a lift using your previous remarks. | |
| Sep 19, 2018 at 11:57 | comment | added | S. Douteau | And for the obstruction to have a retract, I think that $\text{sk}_1\text{Sd}^1\Delta^2$ already gives a counter example, since the image of the "middle" vertex would have to be connected to every other vertices in $\text{sk}_1\text{Sd}^1\partial\Delta^2$ | |
| Sep 19, 2018 at 11:54 | comment | added | S. Douteau | I did not realize it was enough to have a retract, nor that $|sk_nSd^k∂Δ^{m+1}|→|sknSd^kΔ^{m+1}|$ actually admited one, thank you for pointing it out. I am not sure to see why $ \text{Ex}^{\infty}$ should be a retract of $\text{Sing}(|X|)$. The fact that they are homotopically equivalent is straightforward, but I don't think this implies that one is the retract of the other. I do agree that if it is true, this implies the existence of a lift. | |
| Sep 18, 2018 at 20:25 | comment | added | Tim Campion | However, I believe it is the case that these lifts exist! To see this, use the fact that $Ex^\infty X$ is a retract of $Sing |X|$, so we may replace $Ex^\infty X$ with $Sing |X|$ in the lifting problem. Then by adjunction, it suffices to observe that $|sk_n Sd^k \partial \Delta^{m+1}| \to |sk_n Sd^k \Delta^{m+1}|$ has a retraction. | |
| Sep 18, 2018 at 20:25 | comment | added | Tim Campion | Thanks, Sylvain! I think I was thinking that $sk_n Sd^k \partial \Delta^{m+1}$ was a retract of $sk_n Sd^k \Delta^{m+1}$, just not a deformation retract. This is true after taking geometric realization, but on reflection it's simply not true before geometric realization. It's kind of fun to draw out $sk_1 Sd^2 \Delta^2$ an think about why this is so. The obstruction has a sort of "metric" feel. | |
| Sep 18, 2018 at 12:00 | review | Late answers | |||
| Sep 18, 2018 at 12:14 | |||||
| Sep 18, 2018 at 11:43 | history | answered | S. Douteau | CC BY-SA 4.0 |