Timeline for Topological groupoids and equivariant sheaves
Current License: CC BY-SA 4.0
22 events
| when toggle format | what | by | license | comment | |
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| Jan 20 at 10:22 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
replaced the dead link
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| Mar 4, 2023 at 8:29 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
Included the title of the linked paper
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| Mar 21, 2022 at 23:44 | answer | added | David Roberts♦ | timeline score: 3 | |
| Mar 21, 2022 at 12:48 | comment | added | user478652 | Note: I removed one question from that post and outsourced it here: mathoverflow.net/questions/418590/skeletal-topological-groupoid | |
| Mar 21, 2022 at 12:42 | history | edited | user478652 | CC BY-SA 4.0 |
deleted 255 characters in body
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| Mar 21, 2022 at 12:41 | vote | accept | user478652 | ||
| Mar 21, 2022 at 11:23 | comment | added | user478652 | @KonradWaldorf I don't see which of my questions Section 3.2 answers. | |
| Mar 16, 2022 at 16:59 | comment | added | Zhen Lin | Yes, I am referring to the notion of equivalence you defined in Q1. Other notions considered in this context are weaker, so this one is "strong". | |
| Mar 16, 2022 at 11:28 | comment | added | user478652 | @ZhenLin Thank you very much! What do you mean by "(strongly) equivalent" in your first comment? The notion I defined in Q1 or some of the other notions of equivalence people mentioned or linked to? | |
| Mar 16, 2022 at 10:47 | comment | added | Zhen Lin | @user478652 If you choose the "same" pre-bound, jointly surjective set of points, and cardinal, then you get isomorphic topological groupoids. I consider this to be obvious but the details are tedious. | |
| Mar 16, 2022 at 10:45 | comment | added | Zhen Lin | @user478652 There is a "canonical" essentially small site: pick the least $\lambda$ such that the topos is $\lambda$-accessible and that the class of $\lambda$-presentable objects is closed under finite limits, then the full subcategory of $\lambda$-presentable objects with an appropriate topology will be an essentially small site. This is not very functorial, however. | |
| Mar 16, 2022 at 9:22 | comment | added | Konrad Waldorf | @user478652: Section 3.2 explains the difference between the morphisms you described and the morphisms Benjamin Steinberg pointed out above. | |
| Mar 16, 2022 at 9:12 | comment | added | user478652 | @KonradWaldorf Clicking on the link I see words like "stack", "orbifold", and "manifold", but nothing related to my question (I'd had to search hard to find it!). Could you point me specifically to the sections in the paper that answer some of my questions? | |
| Mar 16, 2022 at 9:08 | comment | added | Konrad Waldorf | I recommend to read this paper of Metzler: arxiv.org/abs/math/0306176 | |
| Mar 16, 2022 at 9:01 | comment | added | user478652 | @ZhenLin That is, fix an equivalence $\mathcal F\colon\mathcal E\to\mathcal E'$. Let $T$ be a pre-bound for $\mathcal E$. Then choose $\mathcal F(T)$ as the pre-bound for $\mathcal E'$. Let $X$ be a set of enough points of $\mathcal E$. Then let the image of $X$ under $\mathcal F$ be the chosen set of enough points for $\mathcal E'$. | |
| Mar 16, 2022 at 8:57 | comment | added | user478652 | @ZhenLin Concerning Q5: my question is meant in the following sense: if $\mathcal E$ and $\mathcal E'$ are equivalent and we choose the "same" pre-bound, jointly surjective set of points, and cardinal for both $\mathcal E$ and $\mathcal E'$, are the result topological groupoids equivalent or even isomorphic? | |
| Mar 16, 2022 at 8:54 | comment | added | user478652 | @ZhenLin Thank you very much! Concerning your answer of Q4: by strongly equivalent topological groupoids you mean the notion I defined in Q1, right? The topos itself is not a site (using my definition of site), because it is not small. Is there a canonical small site, similar to Example 2.1.11(c) in the Elephant? | |
| Mar 16, 2022 at 7:59 | comment | added | Zhen Lin | Regarding Q5: It occurs to me that there is not even "the" Butz–Moerdijk groupoid associated to one topos. There are a number of arbitrary choices that have to be made: first, you have to choose a pre-bound, then you have to choose a jointly surjective set of points, and then you have the choose a sufficiently large cardinal. It is not obvious to me that the groupoid you get is independent of these choices in any meaningful sense other than Morita equivalence. | |
| Mar 15, 2022 at 22:20 | comment | added | Zhen Lin | Regarding Q4: Well, the topos itself is a a site. 2-functoriality of the equivariant sheaves construction ensures that (strongly) equivalent topological groupoids yield the same topos. The converse is false due to the Morita equivalence phenomenon. | |
| Mar 15, 2022 at 18:53 | comment | added | Benjamin Steinberg | People often use a different definition of equivalence for topological groupoids because (1) is false. See for example mathoverflow.net/questions/111801/… | |
| Mar 15, 2022 at 18:51 | answer | added | Benjamin Steinberg | timeline score: 6 | |
| Mar 15, 2022 at 17:27 | history | asked | user478652 | CC BY-SA 4.0 |