Timeline for $(\infty,1)$-topoi generated by $(n,1)$-categories
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 25, 2023 at 11:59 | comment | added | D.-C. Cisinski | What I meant is that the theory of $n$-hyperlocalic $\infty$-topoi is easier to understand because it behaves more like the theory of $n$-topoi (because $n$-topoi for finite $n$ are hypercomplete), in particular when it comes to finding sites of definition. | |
| Jan 25, 2023 at 11:59 | comment | added | D.-C. Cisinski | $n$-localic topoi are not hypercomplete, but we can still define "$n$-hyperlocalic topoi" as those $\infty$-topoi that are both hypercomplete and generated by their $(n-1)$-truncated objects (these are not $n$-localic topoi in the sense of HTT though). Then the operator $\tau_{\leq n-1}$ induces an equivalence from $n$-hyperlocalic topoi to $n$-topoi. This also implies that the assignment $X\mapsto X^\wedge$ defines an equivalence from $n$-localic topoi to $n$-hyperlocalic ones (that is far from being the identity, though). | |
| Jan 25, 2023 at 4:59 | comment | added | Mike Shulman | One interesting observation about these "weakly $m$-localic" topoi is that (I think) the collection of them all is an $(m,2)$-category, like the collection of $m$-localic topoi. | |
| Jan 25, 2023 at 4:57 | history | edited | Mike Shulman | CC BY-SA 4.0 |
added assumption of finite limits
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| Jan 25, 2023 at 4:53 | comment | added | Mike Shulman | @D.-C.Cisinski I still don't understand what you're saying. I said that an $n$-localic topos may NOT be hypercomplete. Rezk's example cited by Marc shows this. | |
| Jan 25, 2023 at 4:53 | comment | added | Mike Shulman | @MarcHoyois Thanks, I see! | |
| Jan 24, 2023 at 19:02 | comment | added | Marc Hoyois | If $Shv(X)^{hyp}$ is $n$-localic then it must agree with $Shv(X)$ since they have the same $(n-1)$-truncated objects and the latter is $0$-localic. But in Rezk's example $Shv(X)$ is not hypercomplete. | |
| Jan 24, 2023 at 7:01 | comment | added | D.-C. Cisinski | Yes, localic $n$-topoi need to be hypercomplete. What I meant is that we can characterize $n$-localic $\infty$-topoi as those hypercomplete $\infty$-topoi that are generated by their $(n-1)$-truncated objects (but a general $\infty$-topos generated by its truncated objects need not be hypercomplete). I also meant that Marc's concerns about having a site with finite limits are not really important because we restrict ourselves to hypercomplete topoi anyway. | |
| Jan 23, 2023 at 18:51 | comment | added | Mike Shulman | @D.-C.Cisinski What do you mean by "restrict ourselves to hypercomplete $(\infty,1)$-topoi"? Certainly an $n$-localic topos need not be hypercomplete? | |
| Jan 23, 2023 at 18:50 | comment | added | Mike Shulman | @MarcHoyois How do you conclude that it can't be $n$-localic? | |
| Jan 23, 2023 at 16:46 | comment | added | Marc Hoyois | It seems that it fails even when $n=0$: there is an example of Charles Rezk of a topological space whose hypercomplete sheaves are presheaves on a poset, so that the latter cannot be $n$-localic for any finite $n$. So it seems "flatness" for functors to $\infty$-topoi is strictly stronger than for $n$-topoi, which is similar to what happens with "dense" subsites. | |
| Jan 23, 2023 at 10:25 | comment | added | D.-C. Cisinski | Since $(n-1)$-truncated objects are stable under finite limits, if we restrict ourselves to hypercomplete $(\infty,1)$-topoi, it looks like being $n$-localic really means being an hypercomplete left exact localization of a category of presheaves on an $(n,1)$-category. In particular, presheaves of $\infty$-groupoids on a $(n,1)$-category form a hypercomplete $(\infty,1)$-topos generated by its $(n-1)$-truncated objects and thus define a $n$-localic topos. | |
| Jan 22, 2023 at 17:49 | comment | added | Mike Shulman | If it's not true, then I would also want to assume finite limits for the class I'm interested in. | |
| Jan 22, 2023 at 17:48 | comment | added | Mike Shulman | @MarcHoyois Good question. It's true when $n=0$, right? I would be surprised if it weren't true; I assumed the assumption of finite limits was just to avoid talking about flat functors. | |
| Jan 22, 2023 at 8:56 | comment | added | Marc Hoyois | Is it really true that presheaves on a $n$-category form an $n$-localic $\infty$-topos? The statement in HTT assumes that the $n$-category has finite limits. | |
| Jan 22, 2023 at 1:44 | history | asked | Mike Shulman | CC BY-SA 4.0 |