Timeline for Commutative rings : Topoi = Fields :?
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21 events
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| Feb 10 at 5:36 | comment | added | user30211 | @MikeShulman it seems that the presentable condition allows for a certain kind of tensor product. This tensor can express a universal property for spectra. | |
| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| S Jun 10, 2019 at 20:13 | history | suggested | Victoria M | CC BY-SA 4.0 |
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| Jun 10, 2019 at 19:53 | review | Suggested edits | |||
| S Jun 10, 2019 at 20:13 | |||||
| Jun 10, 2019 at 19:00 | answer | added | Tim Campion | timeline score: 5 | |
| Feb 16, 2019 at 21:41 | answer | added | Qiaochu Yuan | timeline score: 10 | |
| Feb 16, 2019 at 11:58 | comment | added | Mike Shulman | But it seems that general monoidal suplattices are rather closer to frames than general monoidal locally presentable categories are to toposes. In particular, a monoidal suplattice whose monoidal structure is cartesian is already a frame, whereas the same is not true in the categorified case because the monoidal structure can only be a cartesian product rather than a pullback. | |
| Feb 16, 2019 at 11:56 | comment | added | Mike Shulman | Personally, I think this analogy is not quite right as stated. A better analogy to rings is monoidal (locally) presentable ($\infty$-)categories, since there is nothing in the notion of "ring" implying that the multiplication should be a "cartesian product". With this modification, both in fact become instances of monoids in a monoidal category that is "additive" in some sense, and the analogy gets even stronger when looking at the intermediate notion of monoids in suplattices, which include frames/locales as special cases. | |
| Feb 15, 2019 at 21:38 | comment | added | Phil Tosteson | @Ivan I think you're right, I guess I need to refresh my abstract algebra | |
| Feb 15, 2019 at 21:09 | comment | added | Simon Henry | Not that I remember. | |
| Feb 15, 2019 at 21:01 | comment | added | Ivan Di Liberti | I don't really agree, those could be, at most, the analogue of integral domains. On the other hand, the perspective is a good one, @SimonHenry, did you discuss also domains with Mathieu? | |
| Feb 15, 2019 at 15:27 | comment | added | Phil Tosteson | Topoi with the property that $$X \times Y = \emptyset \implies (X = \emptyset) \vee (Y = \emptyset),$$ seem like a reasonable candidate. | |
| Feb 15, 2019 at 9:22 | comment | added | Ivan Di Liberti | "'Topos' are "affine scheme", the correspondence between them corresponds the "Spec" "Global section". Hence by definition Set/Space is the only logos whose 'spectrum' is a point." might be the good answer I was looking for. It also shows that topoi are $\textit{some}$ commutative rings. | |
| Feb 15, 2019 at 9:13 | comment | added | Simon Henry | I remember having this conversation with Mathieu... but not its exact content. We couldn't give a precise and definite answer to this question, but all our informal argument were pointing to the fact that there is only one 'field', topos of spaces/sets. One of them was that in this analogy, 'Logos' (as Joyal call them, i.e. topos with $f^*$ morphisms between them) are ring, 'Topos' are "affine scheme", the correspondance between them corresponds the "Spec" "Global section". Hence by definition Set/Space is the only logos whose 'spectrum' is a point. | |
| Feb 15, 2019 at 1:31 | comment | added | David Roberts♦ | Joyal made a similar point about (1-)toposes and rings at Topos à l'IHÉS, coming at it via frames v locales as 0-toposes. It would be in the videos that are on YouTube, unfortunately I don't know which of his lectures it's in, or at what point. | |
| Feb 15, 2019 at 0:08 | comment | added | Theo Johnson-Freyd | If you do get an answer in exchange for beer, please share it (the answer, and the beer if you can figure out how) with the rest of us! | |
| Feb 14, 2019 at 23:08 | comment | added | fosco | mmmh, nice question! I see an obstruction in making a reasonable categorification in the fact that fields are not algebraic. Also, why don't you ask Mathieu (hopefully, to have a similar diagram in exchange for Czech beer)? :-) | |
| Feb 14, 2019 at 22:42 | comment | added | Andrej Bauer | Please don't post pictures of text in your questions. The text in them is not searchable (yet). | |
| Feb 14, 2019 at 22:33 | history | edited | Ivan Di Liberti | CC BY-SA 4.0 |
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| Feb 14, 2019 at 21:42 | history | edited | Ivan Di Liberti | CC BY-SA 4.0 |
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| Feb 14, 2019 at 20:56 | history | asked | Ivan Di Liberti | CC BY-SA 4.0 |