Timeline for Every Grothendieck topos can be built from localic topoi
Current License: CC BY-SA 4.0
24 events
| when toggle format | what | by | license | comment | |
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| Feb 26, 2022 at 22:27 | comment | added | Zhen Lin | @user477332 I was referring to the language/nationality. | |
| Feb 26, 2022 at 17:04 | vote | accept | user477332 | ||
| Feb 26, 2022 at 17:04 | comment | added | user477332 | @TimCampion I see. My main questions are about making precise the things you sketched, because I don't have much experience in higher category theory (while you answer as an expert in higher category theory). I agree that the comment section might not be the best place for long discussions. So maybe I should just keep learning higher category theory and come back to this thread in a while to see whether I can prove the things myself! Let me say thank you (and also thanks to Simon Henry and Zhen Lin Low) - your answer(s) already gave me an idea of what's going on. :-) | |
| Feb 26, 2022 at 16:12 | comment | added | Tim Campion | @user477332 i take that back. Rather, let me reassure you that it’s quite normal for two mathematicians to realize partway through a conversation that they don’t share the same background knowledge. No need to get worked up over it. I’m having difficulty gauging where our common language does lie, though. So in order to help I do need it explained to me what is confusing at this point. | |
| Feb 26, 2022 at 15:45 | comment | added | Tim Campion | @user477332 I sense some frustration. I think you're expecting a back-and-forth where we peel back the layers together, trying to identify exactly what you know and what you don't know in order to tailor an answer to your particular existing knowledge, like a teacher in office hours. MO is not ideally suited to such a process (it's not a chat platform) -- it's more like bumping into a colleague at tea and chatting for a few minutes. That said, several people have been explaining things, and I've lost track of what you're still confused about. If you can catch me up, I can try to help. | |
| Feb 26, 2022 at 13:54 | comment | added | user477332 | @TimCampion There are still a lot of questions left ... I won't accept your answer until you give either rigorous proofs of all statements or precise references for all the claims, rather than an answer that just experts are able to understand using a lot of folklore knowledge that is not available to me. | |
| Feb 26, 2022 at 13:53 | comment | added | user477332 | @SimonHenry Thanks for answering some of my questions. | |
| Feb 26, 2022 at 13:52 | comment | added | user477332 | @ZhenLin No, the Polish spelling is "Čech" too: pl.wikipedia.org/wiki/Eduard_%C4%8Cech. | |
| Feb 20, 2022 at 7:51 | comment | added | Zhen Lin | Fwiw, "Čech" is the Czech spelling and "Czech" is the Polish spelling. | |
| Feb 19, 2022 at 19:29 | comment | added | Simon Henry | The fact that equivariant sheaves corresponds to the colimits of the simplicial diagram follows from two fact : 1) colimits of Grothendieck toposes are computed as (pseudo)-limits of the underlying categories (for the f^* functors). 2) The definition of an equivariant sheaves is exactly the spelled out version of what this pseudo-limits is for a 2-truncated simplicial object.... Oh And yes the "Czech nerve" was indeed just a typo, but that's kinda funny because Cech is Czech, so I guess it is indeed a Czech nerve. | |
| Feb 19, 2022 at 19:24 | comment | added | Simon Henry | @user477332. Not all (truncated or not) simplicial objects corresponds to a groupoids objects, but groupoids objects form a full subcategory of (truncated) simplicial objects they can be characterized by so called Segal conditions. WHat I said about being in a 2-category is because if you take the colimits of a simplicial objets in a 1-category the results only depends on the first two object, in a 2-category it depends on the first three objects (and more generally in an n-category it depends on the first (n+1)-objects.) | |
| Feb 19, 2022 at 16:53 | comment | added | user477332 | What's funny about "Czech nerve"? Seems like a misspelling? | |
| Feb 19, 2022 at 16:51 | comment | added | user477332 | @SimonHenry Yes, we are doing higher category theory on the meta level, because we work with the 2-category of topoi. But I was referring to the fact Tim talked about "-cells" of that simplicial diagram. That's on another stage. But now I understand what he means by "-cells". But what do you mean by "that's why you need to go one level higher in the simplicial complexe than you would normally do in a 1-category"? Even if you talk about internal categories in a 1-categories I think you need 2-cells to encode composition of you want to code the internal category as a simplicial object. | |
| Feb 19, 2022 at 16:45 | comment | added | user477332 | Also, nLab talks about internal categories in a 1-category. But we are in a 2-category (the 2-category of topoi). | |
| Feb 19, 2022 at 16:42 | comment | added | user477332 | I don't have 10 years of experience with higher categories - you have to make these claims precise if you want that I understand what you mean. :-) People often give nLab links but mostly I find them unhelpful because I have the feeling only experts can understand them. :-( | |
| Feb 19, 2022 at 16:41 | comment | added | user477332 | "You can more generally think of an internal category object (not just an internal groupoid object) in a category as a certain type of simplicial object, as described here." Why? The nLab link you gave descibes an internal nerve construction. Are you claiming that this construction exhibits a bijection between internal categories in A and 2-truncated simplicial objects $S_\bullet$ in A satisfying the condition that each $\Lambda_1^2\to S_\bullet$ has a unique filler? (Is that what you mean by "a certain type of simplicial object" or am I missing a condition?) | |
| Feb 19, 2022 at 16:36 | comment | added | user477332 | There are many things that are unclear to me. Why is a localic groupoid the same as such a 2-truncated simplicial diagram in the 2-category of topoi? Why does the topos of equivarient sheaves of a localic groupoid coincide with the colimit of such a 2-truncated simplicial diagram? What do you mean by the statement that the axioms are automatically satisfied when one takes pseudopullbacks, and how do you prove that? How do you know that there are implicitly also some maps going back in the reverse direction? Lurie didn't write that. | |
| Feb 19, 2022 at 15:23 | comment | added | Tim Campion | @SimonHenry haha "Czech nerve" :) | |
| Feb 18, 2022 at 17:41 | comment | added | Simon Henry | Also, you are doing higher category here : you are working in the $2$-category of toposes, that's why you need to go one level higher in the simplicial complexe than you would normally do in a $1$-category. | |
| Feb 18, 2022 at 17:40 | comment | added | Simon Henry | @user477332 : In general for a map $U \to X$ in a category, its Czech nerve is the simplicial complex whose $n$-th object is the $(n+1)$-folds fiber product $U \times_X U \times_X \dots \times_X U$. The object Tim is talking about corresponds to the first three objects of that complex. In this case you need $2$-cells to get the groupoids structure: the associativity of the groupoid multiplication is expressed as a certain equation between maps $U \times_X U \times_X U \to U $, which is part of the simplicial structure at level $2$. | |
| Feb 18, 2022 at 16:34 | history | edited | Tim Campion | CC BY-SA 4.0 |
added 1107 characters in body
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| Feb 18, 2022 at 16:30 | comment | added | user477332 | "it has topos of objects U, topos of 1-cells U×XU, and so forth" this suggests there are 2-cells. but what do you mean by n-cells and why do we need 2-cells? we are not doing higher category theory here! | |
| Feb 18, 2022 at 16:28 | comment | added | user477332 | Thanks. I don't see any "truncated simplicial diagram", though. I also don't know what that is. :-) | |
| Feb 18, 2022 at 16:26 | history | answered | Tim Campion | CC BY-SA 4.0 |