Timeline for What interesting/nontrivial results in Algebraic geometry require the existence of universes?
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26 events
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| Jun 1, 2021 at 10:43 | history | edited | Joel David Hamkins |
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Jul 31, 2010 at 16:59 | answer | added | Andreas Blass | timeline score: 22 | |
| Jun 21, 2010 at 8:41 | vote | accept | Harry Gindi | ||
| Jun 21, 2010 at 4:25 | answer | added | Joel David Hamkins | timeline score: 54 | |
| Jun 16, 2010 at 15:34 | comment | added | Harry Gindi | Nope, but I'm sure that François G. Dorais or Joel David Hamkins could, if you were really interested. | |
| Jun 16, 2010 at 15:11 | comment | added | Pierre-Yves Gaillard | Wow! Do you have a reference? | |
| Jun 16, 2010 at 14:48 | comment | added | Harry Gindi | I used to be a partisan for Bourbaki's set theory, but there are ways of recasting a stronger version of Bourbaki's set theory as ZFC+$\varepsilon$+Universes, which is more standard but just as rigorous. In particular, since Bourbaki does not contain the axiom of replacement, it is a stronger version of Zermelo+Choice(sometimes +universes). | |
| Jun 16, 2010 at 13:46 | comment | added | Pierre-Yves Gaillard | Dear Harry: Thank you very much for your answer, which confirms my intuition. Would it be indiscreet to ask: What is your personal position on these questions? | |
| Jun 16, 2010 at 11:55 | comment | added | Harry Gindi | Not to my knowledge, no. Mainly because the first is built into the $\tau$ operator, the second is not equivalent to Bourbaki's set theory, and the third because there is no class theory in Bourbaki, since sets are instantiated by the $\tau$ operator. | |
| Jun 16, 2010 at 11:41 | comment | added | Pierre-Yves Gaillard | Has Grothendieck ever used expressions such as "Axiom of Choice", "Zermelo-Fraenkel", "proper class", etc? | |
| Jun 16, 2010 at 11:32 | comment | added | Lars | J. de Jong's blog entry from today seems to be relevant to this discussion: math.columbia.edu/~dejong/wordpress/?p=530 | |
| Jun 16, 2010 at 9:12 | comment | added | Pierre-Yves Gaillard | In my humble opinion, Grothendieck's main motivation for introducing the Axiom of Universes was to make Category Theory Bourbaki compatible. Actually, this is implicit in Harry Gindi's statement of the question. Harry writes "this relative approach makes proper classes pointless". But from Bourbaki's perspective, the expression "proper class" just doesn't make sense. In short (in my interpretation of Grothendieck's thought) the function of this axiom was to make Category Theory soundly founded. | |
| Jun 16, 2010 at 7:26 | history | edited | Harry Gindi | CC BY-SA 2.5 |
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| May 15, 2010 at 4:28 | comment | added | BCnrd | @Kevin: I have no recollection of making such a statement; maybe I said something like not being able to see why such a notion is universe-independent (as among reasons why I never accepted the universe stuff as valid for myself). Anyway, I've never encountered a situation where an fpqc cohomology group intervenes in a relevant way (this is distinct from the perfectly useful and concrete notions that certain functors may satisfy sheaf axioms for fpqc covers or that some object is an fpqc-torsor for some fpqc group scheme). | |
| May 14, 2010 at 21:09 | comment | added | Kevin Buzzard | @BCnrd: I think you once told me (perhaps in the 90s) that certain fpqc cohomology groups were universe-dependent (i.e. the answer depended on what universe you were in). Did I remember correctly? If I did then perhaps these would serve as examples. | |
| May 14, 2010 at 15:11 | comment | added | BCnrd | @Torsten: with a little practice, the "distance between intuition and reality" is also short when attentive to bounding issue: when I encounter a new topos that I have reason to care about, I go through a couple of exercises in my mind, always quick and painless (not tedious or cumbersome). The far-out things to give up (e.g., Hom's between presheaves on big etale site being a set, sheafification on fpqc site) aren't needed for anything I care about, so seems to work fine (and keeps certain abstract things more grounded in reality) for me. In the end, everyone chooses their own poison. :) | |
| May 14, 2010 at 8:11 | comment | added | Torsten Ekedahl | @BCnrd: It may very well be a question of style. Personally, I prefer to get an intuitive picture and worry about details when I have to (of course when I do have to it often enhances my intuition). Having étale toposes lying above the category of schemes is to me an intuitively satisfying picture irrespective of whether or not I throw in universes. If I actually want to go beyond intuition I know that I have to throw in universes. This makes the distance between intuition and mathematical reality shorter than if I have to worry about bounding everything so as to avoid universes. | |
| May 14, 2010 at 6:02 | comment | added | BCnrd | @Torsten: for etale or fppf topologies, doesn't seem to use much time to bypass universes (hard parts of theory remain what takes time to absorb), and I can't think of results whose formulations are harder to understand (e.g., no need to restrict the schemes one uses, etc.) But may be matter of style, since my own preference with constructions is to see in "hands-on" terms what makes it work (and appeal to universes seems to mask that; universes can't be reason something "real" works). Likewise, fpqc sheaf makes sense without "general theory" of fpqc topos, or "Hom-set" between such sheaves. | |
| May 14, 2010 at 4:33 | comment | added | Torsten Ekedahl | I would say it is very much like the axiom of choice; universes can mostly be avoided (even more so than in the case of AC) but it would force one to spend time on the least interesting parts of the theory and results would have to be qualified so as to make them more difficult to understand. I doubt very much that there are interesting results which would not allow such a reformulation. It is however very convenient when one considers things like the fibered category of étale toposes over the category of schemes. | |
| May 14, 2010 at 2:16 | history | edited | Harry Gindi | CC BY-SA 2.5 |
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| May 14, 2010 at 0:30 | history | edited | Harry Gindi | CC BY-SA 2.5 |
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| May 14, 2010 at 0:27 | comment | added | Harry Gindi | Here's the actual quote, which is a bit more nuanced than I remembered, but the idea there is pretty similar, "The suggested references appeal to fpqc-sheafifiction, a somewhat far-out operation. (I generally view appeal to universes as a bit of laziness when it can be avoided with a bit more effort to unravel what is actually going on.)" | |
| May 13, 2010 at 23:26 | comment | added | Harry Gindi | Because that's the theorem that everyone cites when explaining universes for the first time. | |
| May 13, 2010 at 23:15 | comment | added | Mariano Suárez-Álvarez | Why "of course"? | |
| May 13, 2010 at 23:11 | history | asked | Harry Gindi | CC BY-SA 2.5 |