Timeline for Necessary and sufficient conditions for all sheaves on a site to be continuous functors?
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Dec 19, 2022 at 6:37 | vote | accept | Arshak Aivazian | ||
| Dec 15, 2022 at 9:11 | answer | added | Simon Henry | timeline score: 4 | |
| Dec 15, 2022 at 8:42 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| Dec 15, 2022 at 8:38 | comment | added | Arshak Aivazian | As I wrote in the comments above, I'm more asking about ways to establish a subcanonical site structure on a small category so that all sheaves are continuous functors. Implementing all continuous functors as sheaves is more of a nice bonus (and your comments make me think it's hard to achieve). I'll edit the question to match those emphases, thanks. | |
| Dec 15, 2022 at 8:33 | comment | added | Simon Henry | By "strange question" what I mean is: you are very probably not asking what you actually want to know, you should tell use more about what it is you are trying to do, because that specific question is hard for reason that are really not interesting. | |
| Dec 15, 2022 at 8:30 | comment | added | Simon Henry | For any topos $T$ you can take $T$ with its canonical topology and then sheaves = continuous presheaf = representable presheaf. But If you want a small site $C$ where sheaf are all continuous this is is a strange question because a small category (which isn't a poset) never has all limits so this is about finding a generating subcategory of the topos that isn't going to have colimits that aren't colimits in the topos... | |
| Dec 15, 2022 at 8:26 | comment | added | Arshak Aivazian | Yes, I mean all small limits. | |
| Dec 15, 2022 at 8:22 | comment | added | Simon Henry | Also you should clarify how you want to deal with size problems (when you say all limits, do you mean small limits ?) | |
| Dec 15, 2022 at 8:22 | comment | added | Simon Henry | There are quite a lot of functor that preserves all limit (even the non-small one) Any right adjoints functor ? To me the problem is more that asking for limits to be preserved when limit don't exists is strange - you may want to look at flat functors, or functor that preserves some class of limits that exists - or some specific set of marked limits, and then you have nice answer to your question. But asking it for "all limits that exists" is strange. For the second question - it is the same to ask that the embedding of the site into the topos preserves colimits. | |
| Dec 15, 2022 at 7:51 | comment | added | Zhen Lin | Sure. It was just an example. My point is that it is very difficult to say anything about "all limits". Can you give any non-trivial example of a functor that preserves "all limits" – other than representable functors, adjoints, or things derived from those? | |
| Dec 15, 2022 at 7:21 | comment | added | Arshak Aivazian | @ZhenLin The limit of a functor whose index diagram has an initial object is in effect always equal to the value of that functor on the initial object. Therefore, this condition is trivially satisfied, right? | |
| Dec 15, 2022 at 7:07 | comment | added | Zhen Lin | @AivazianArshak That definition is wild (= badly behaved, uncontrolled). For example, if $I$ has a terminal object then $\textrm{id} : I^\textrm{op} \to I^\textrm{op}$ has a limit, so you would demand that $F : I^\textrm{op} \to \textbf{Set}$ preserves it. | |
| Dec 15, 2022 at 4:41 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| Dec 15, 2022 at 4:37 | comment | added | Arshak Aivazian | Is it possible that almost every small category is realized as a category of k-representable objects? If so, then my first (stronger) question was already asked on the site. But in fact, I'm more interested in topoi all of whose sheaves are continuous than in topoi of all continuous sheaves. What about my second question? | |
| Dec 15, 2022 at 3:19 | comment | added | Arshak Aivazian | @ZhenLin I mean: functor $F : I^{op} \to \mathrm{Set}$ is continuous iff $F$ preserves all the limits that $I^{op}$ has. | |
| Dec 15, 2022 at 3:11 | comment | added | Zhen Lin | Can you be more precise about how you define “continuous”? Particularly in the case where the site does not have “all” colimits. | |
| Dec 15, 2022 at 3:00 | history | asked | Arshak Aivazian | CC BY-SA 4.0 |