Timeline for On Street's "australian conspectus"
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 31, 2018 at 21:19 | comment | added | Mike Shulman | By the way, this is really three unrelated questions. I think it would be better to ask them all separately. | |
| Jan 31, 2018 at 21:18 | comment | added | Mike Shulman | One thing worth noting is that while the associated $n$-sheaf functor can be written as $L^{n+2}$, this fails for $n=\infty$ because the induction in the finite case is "downwards". That is, when acting on an $n$-presheaf, $L$ first makes the top dimension sheafy, then the next-to-top dimension, and so on, so with an $\infty$-presheaf there is "nowhere to get started". (If instead the induction worked upwards, starting from the bottom dimension, then one might expect that $\omega$ applications of $L$ would produce $\infty$-sheafification.) | |
| Jan 29, 2018 at 20:21 | comment | added | David Roberts♦ | If you consider presheaves valued in the poset $0\to1$, then one application of $L$ should be enough to give a sheaf. You should think of this poset as being the universe Prop of propositions aka (-1)-types. | |
| Jan 29, 2018 at 18:00 | answer | added | Tim Porter | timeline score: 5 | |
| Jan 29, 2018 at 10:34 | comment | added | მამუკა ჯიბლაძე | I absolutely agree that 3. is amazing, underrated and very poorly understood. Details are all hidden in the proof of 3.8 of his "Two-dimensional sheaf theory" | |
| Jan 29, 2018 at 9:50 | comment | added | Denis Nardin | Uh $n+1$ should have been $n+2$ in my comment above... | |
| Jan 29, 2018 at 9:37 | answer | added | Simon Henry | timeline score: 7 | |
| Jan 29, 2018 at 9:22 | comment | added | Simon Henry | That is indeed exactly what happen in general. But I'm not sure what Fosco is asking exactly for this thrid question. Could you clarify the sort of things you want Fosco ? Are you also looking for an explanation of why the localization is "quadratic" for sets ? Or is just why is it $L^2$ for sets and $L^3$ for 1-stack ? | |
| Jan 29, 2018 at 9:14 | comment | added | Denis Nardin | For 3., I suspect that (talking in homotopy-theory speak) for a presheaf $P$ of spaces an application of $L$ decreases the "truncatedness" of the map $P(X)\to \lim P(\check{U})$ (so if our presheaf were a presheaf of $n$-truncated spaces to begin with, $(n+1)$-applications of $L$ turn it into a sheaf). | |
| Jan 29, 2018 at 8:29 | history | asked | fosco | CC BY-SA 3.0 |