Timeline for The Barr-Boole-Galois topos; a modification of sets to play well with schemes
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Nov 10, 2021 at 22:42 | comment | added | Mendieta | I don't think it answers the question but the paper cahierstgdc.com/wp-content/uploads/2017/05/Menni_55-2.pdf seems related. | |
| Dec 20, 2019 at 1:27 | comment | added | David Roberts♦ | Yes, something like cohesion, I guess. In the condensed world, the category of sheaves on a point (Set, in the usual setup) is the proétale site of the point, which is the weird big thing that contains all the Stone spaces etc. | |
| Dec 19, 2019 at 19:27 | comment | added | user30211 | Maybe schemes to condensed schemes is a cohesion setup - is that what you were suggesting? Of course, we already have a geometric morphism between schemes and condensed schemes. I think I'll check for further adjoints on each side. | |
| Dec 19, 2019 at 19:26 | comment | added | user30211 | @DavidRoberts can you elaborate a bit, maybe with analogies? The Barr-Boole-Galois topos is to Schemes as what is to what, in condensed mathematics? | |
| Dec 16, 2019 at 10:47 | comment | added | David Roberts♦ | Question 2) feels a little bit like a baby version of Clausen and Scholze's condensed mathematics, where the base is changed from set to something a bit more...exotic. | |
| Dec 16, 2019 at 9:52 | comment | added | Todd Trimble | I'm not aware of any such modification, and I kind of doubt it since you'd still need conditions on monads/comonads on $\mathrm{Top}$ that come from the string of adjunctions. | |
| Dec 16, 2019 at 7:40 | history | edited | user30211 | CC BY-SA 4.0 |
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| Dec 16, 2019 at 5:02 | comment | added | user30211 | @ToddTrimble. Thanks, I'll fix shortly. Or alternatively, could one modify Set instead of modifying Top? | |
| Dec 16, 2019 at 0:48 | comment | added | Todd Trimble | You gotta be a little careful with the go-to example: It is evident that the discrete space functor $\mathrm{Set} \to \mathrm{Top}$ does not preserve products, because for example Cantor space $2^\mathbb{N}$ is not discrete. You probably want, instead of $\mathrm{Top}$, the category of locally connected spaces, where for example the product topology is refined by applying coreflection $\mathrm{Top} \to \mathrm{LocConn}$ (retopologize by letting connected components of opens be open). | |
| Dec 16, 2019 at 0:08 | comment | added | David Roberts♦ | It could be something related to the Barr cover/Barr embedding theorem. Looking through Barr's topos theory papers I can't see anything obvious (Mike Barr keeps all his papers/books on his website to make them freely accessible) | |
| Dec 16, 2019 at 0:07 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
Put quote in text, minor TeX edits, added link to full article
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| Dec 15, 2019 at 23:24 | history | edited | user30211 | CC BY-SA 4.0 |
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| Dec 15, 2019 at 23:18 | history | edited | user30211 | CC BY-SA 4.0 |
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| Dec 15, 2019 at 23:15 | comment | added | Simon Henry | "Boolean part" is not really a standard terminology, so I can't guarantee that what follows is correct without a reference to something written by Lawvere about this. But if it is a boolean subtopos, I would assume it is the double negation subtopos. (ncatlab.org/nlab/show/double+negation#topology) | |
| Dec 15, 2019 at 23:03 | history | asked | user30211 | CC BY-SA 4.0 |