Timeline for The universal property of the unseparated derived category
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Oct 10, 2017 at 1:24 | comment | added | Jacob Lurie | @Yonatan It's not part of the definition, but it does happen automatically (every topos can be realized as the category of sheaves with respect to the canonical topology on itself). | |
| Oct 9, 2017 at 11:15 | comment | added | Yonatan Harpaz | Right, I see what you mean about hypercoverings. But what about filtered colimits? Shouldn't we take only sheaves which send filtered colimits to limits (or is this built into the definition of the canonical Grothendieck topology on ${\cal X}$)? | |
| Oct 9, 2017 at 0:30 | comment | added | Jacob Lurie | @Yonatan I would say that the $\infty$-topos analogue of the unseparated derived category is the construction which carries a $1$-topos $\mathcal{X}$ to the $\infty$-category of space-valued sheaves on $\mathcal{X}$, where you equip $\mathcal{X}$ with the canonical Grothendieck topology. So the universal property is: coproducts go to coproducts, and Cech nerves of effective epimorphisms go to colimit diagrams. (If you ask this for general hypercoverings, you get the $\infty$-topos of hypercomplete sheaves. This is more like the analogue of the usual derived category.) | |
| Oct 8, 2017 at 18:40 | comment | added | Yonatan Harpaz | Thanks! This is indeed cool. I suppose the same thing happens with $\infty$-topoi, no? i.e., if ${\cal X}$ is a topos, or a 1-topos, then I imagine that the $\infty$-topos ${\cal X}_{\infty}$ "generated" from ${\cal X}$ has a universal property both as an $\infty$-topos and as a presentable $\infty$-category: colimit preserving functors from ${\cal X}_{\infty}$ to a given presentable $\infty$-category ${\cal C}$ are the same (?) as functors ${\cal X} \to {\cal C}$ which preserve filtered colimits and send hypercoverings to geometric realizations. | |
| Oct 8, 2017 at 14:55 | comment | added | Jacob Lurie | @Denis: I don't see how say anything about that (except in the case where $\mathcal{C}$ is stable with a nice t-structure, and the functors are required to be right t-exact). | |
| Oct 8, 2017 at 9:09 | vote | accept | Yonatan Harpaz | ||
| Oct 8, 2017 at 8:45 | comment | added | Denis Nardin | This is very cool. Can we derive a universal property for $\mathcal{D}(\mathcal{A})$ out of this? Said differently, can we describe which functors $\mathcal{A}\to \mathcal{C}$ correspond to functors from $\check{\mathcal{D}}(\mathcal{A})$ sending acyclic complexes to 0? | |
| Oct 7, 2017 at 19:10 | history | edited | Jacob Lurie | CC BY-SA 3.0 |
Added some missing subscripts and coproduct-preservation hypotheses that I think are needed when the target category is not prestable.
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| Oct 6, 2017 at 22:07 | history | answered | Jacob Lurie | CC BY-SA 3.0 |