Timeline for Why is category theory the preferred language of advanced algebraic geometry?
Current License: CC BY-SA 4.0
45 events
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| Jul 21, 2023 at 13:19 | review | Close votes | |||
| Jul 24, 2023 at 11:52 | |||||
| Jul 20, 2023 at 13:26 | comment | added | Tom Copeland | @timothyChow, matrix theory received a real shot in the arm with the advent of QM. This rising to an occasion, this synergy informs both subjects. Knowing with whom the developers are arguing also informs--the dialectic. The historical tension between a traditional diff op approach and a more purely grp theoretic one is discussed by Bonolis in "From the Rise of the Group Concept to the Stormy Onset of Group Theory in the New Quantum Mechanics. A saga of the invariant characterization of physical objects, events and theories". | |
| Jul 18, 2023 at 16:54 | comment | added | huurd | @Justin Hilburn : in this later version it is not clear what Grothendieck eventually thought about the situation of Z Mebkhout "I was contacted by Pierre Schapira, and then by Christian Houzel, to point out some glaring inaccuracies in the version of events presented in Récoltes et semailles. The situation was clarified considerably during correspondence with both of them. It now appears to me that in the “[Zoghman] Mebkhout version” (which was not lacking in internal consistency) the true, the tendentious and the downright false are inextricably mixed" | |
| Jul 18, 2023 at 16:03 | comment | added | Justin Hilburn | @huurd See inference-review.com/article/a-truncated-manuscript for Grothendieck's later views on Z Mebkhout. | |
| Jul 18, 2023 at 15:31 | comment | added | Timothy Chow | @huurd I'm not aware of such a link. My go-to reference for this topic is Olver's book Applications of Lie Groups to Differential Equations. | |
| Jul 18, 2023 at 15:24 | answer | added | Donu Arapura | timeline score: 11 | |
| Jul 18, 2023 at 14:17 | comment | added | huurd | @Timothy Chow : is there a link between the use of Lie groups you just mentioned for differential equations, and the theory of differentially closed fields ? en.wikipedia.org/wiki/Differentially_closed_field | |
| Jul 18, 2023 at 13:16 | comment | added | Timothy Chow | @TomCopeland Lie theory was originally motivated in large part by trying to replicate the success of Galois theory in the context of differential equations. As for category theory, its initial development was intertwined with the challenges of understanding algebraic topology, and much of the category theory in algebraic geometry stems from the application of the ideas of algebraic topology to algebraic geometry. | |
| Jul 18, 2023 at 10:27 | comment | added | Paul Taylor | @TomCopeland. (I like the quotations on your profile page.) "category theory has been intertwined with the challenge(s) of understanding . . ." many, many things, because it is an attitude to doing mathematics, not a particular body of it. Try reading the abstracts on the recent international conference. | |
| Jul 17, 2023 at 20:24 | comment | added | Tom Copeland | I'd like to see a possible Toynbee-ish approach. For example, the development of group theory was closely intertwined with the challenge of solving quartic, quintic, and higher-order equations and then with understanding conservation theorems / invariants in mathematical physics; the development of operational calculus, with the challenges of electric circuit theory (and practice); Lie theory, with the challenges of the mathematical underpinnings of quantum theory; and so on. The development of category theory has been intertwined with the challenge(s) of understanding . . . (?). | |
| Jul 17, 2023 at 19:56 | comment | added | Tom Copeland | @AndyPutman, aren't psychologists / psychiatrists typically ensconced in armchairs when practicing their professions? | |
| Jul 17, 2023 at 16:55 | review | Close votes | |||
| Jul 18, 2023 at 9:16 | |||||
| Jul 17, 2023 at 14:13 | comment | added | Timothy Chow | @AndyPutman I'm not sure that it's true that category theory is used more or less the same amount in all algebraic fields. For example, based on my (admittedly superficial) understanding of the classification of finite simple groups, I don't think that it makes heavy use of category theory. In my own field of algebraic combinatorics, category theory does not get used much even in combinatorial representation theory. (Species are category-theoretic but it's a relatively specialized subfield.) But in algebraic geometry, many important definitions require category theory to even state. | |
| Jul 17, 2023 at 13:58 | history | edited | gmvh | CC BY-SA 4.0 |
Corrected spelling
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| Jul 17, 2023 at 12:38 | comment | added | huurd | @ Andy Putman : it turns out that I have been reading those parts of RetS during the last few weeks, and I was precisely wondering to what extent the words of Grothendieck reflect the reality. It's great to hear your opinion about that, but do you think those "paranoid" parts are entirely false, or is there a bit of truth in it ? For example, the unwanted lonliness of Z Mebkhout during a bunch of years while he was discovering important things in the late 70's ? | |
| Jul 17, 2023 at 12:02 | comment | added | Andy Putman | @huurd: No, I don’t think that at all. I don’t think those people actually exist. Everyone that is at all connected to algebraic geometry (including me) acknowledges their massive debt to Grothendieck’s work. You shouldn’t take the more paranoid parts of R&S seriously as an accurate account of the world. Here for instance is an account by one of the people Grothendieck slanders: jmilne.org/math/Documents/GrothendieckandMe.html | |
| Jul 17, 2023 at 10:47 | history | reopened |
Brian Paul Taylor Mikhail Katz Jochen Glueck Carlo Beenakker |
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| Jul 17, 2023 at 9:41 | review | Reopen votes | |||
| Jul 17, 2023 at 10:50 | |||||
| Jul 17, 2023 at 9:05 | history | edited | Brian | CC BY-SA 4.0 |
added 9 characters in body
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| Jul 17, 2023 at 8:45 | history | left closed in review |
Daniele Tampieri Alex M. Alexey Ustinov |
Original close reason(s) were not resolved | |
| Jul 17, 2023 at 8:36 | comment | added | Mikhail Katz | @SBrian, try to clarify your question, and particularly the last sentence. | |
| Jul 16, 2023 at 20:53 | comment | added | Andy Putman | @ElizabethHenning: But can this kind of question really be answered in a reasonable way? I think the answer is really kind of boring: category theory is a useful and expressive language for expressing algebraic concepts, so it is used in basically all algebraic fields. I really don’t think algebraic geometry is any more category-theory intensive than eg representation theory, homological algebra, algebraic topology, etc. | |
| Jul 16, 2023 at 20:31 | comment | added | Elizabeth Henning | This really is not an "opinion-based" question, although it might be borderline "research-level." But isn't the standard here supposed to be analogous to a colleague wandering into your office to ask something? I'll bet a lot of colleagues have had similar questions about why algebraic geometry is so heavy on category theory. | |
| Jul 16, 2023 at 20:17 | comment | added | Andy Putman | @PaulTaylor: I suspect that like this one, such questions would be quickly closed. | |
| Jul 16, 2023 at 20:15 | comment | added | Andy Putman | @huurd: You don’t need the massive baggage of topoi to set up etale cohomology. For instance, this standard source lists them in the index once to point to where he explains that they won’t be used: jmilne.org/math/CourseNotes/LEC.pdf | |
| Jul 16, 2023 at 20:15 | comment | added | Paul Taylor | @AndyPutman: as a categorist, I agree that there's nothing special about algebraic geometry --- aside from the enormous influence that Grothendieck had in an early stage of the development of my subject. There should be parallel Questions with other subjects substituted for AG, each of which deserves a good Answer, all for the Education of the Youth. | |
| Jul 16, 2023 at 20:10 | comment | added | huurd | @Andy Putman : a marginal role ? would you say the same for etale cohomology ? what would etale cohomology be like without the notion of a topos ? | |
| Jul 16, 2023 at 19:47 | comment | added | Andy Putman | @huurd: Topos theory plays a pretty marginal role in mainstream algebraic geometry. I suspect that more algebraic topologists care about it than algebraic geometers. | |
| Jul 16, 2023 at 19:30 | review | Reopen votes | |||
| Jul 17, 2023 at 8:45 | |||||
| Jul 16, 2023 at 19:19 | comment | added | huurd | @Andy Putman : I am not quite sure you're right when you say "There is nothing special about algebraic geometry [and category theory]". For instance, does algebraic topology make use of the notion of topos as does algebraic geometry (refering to Grothendieck's own words in Recoltes et Semailles, it should.. but this was written in the mid 80's, so it might be that meanwhile things have changed) ? Is there an algebraic geometer in the room to confirm my feeling ? I agree with you about the strange comment of S Brian on witchcraft. | |
| Jul 16, 2023 at 18:50 | history | closed |
Andrej Bauer Fernando Muro Neil Strickland Andy Putman abx |
Opinion-based | |
| Jul 16, 2023 at 18:49 | comment | added | Andy Putman | (I also think that the tendentious armchair psychology ("...rather it be seen as witchcraft, perhaps in efforts to bolster the image of their own intelligence") is totally inappropriate.) | |
| Jul 16, 2023 at 18:48 | comment | added | Andy Putman | I think this question is based on a misconception. There is nothing special about algebraic geometry. Category theory is a useful language in almost any algebraic setting, and pretty much anyone that works in algebra uses it to at least some extent. I don't think that algebraic geometry is even the area that uses it the most (that would probably be algebraic topology and homological algebra). There are also important subareas of algebraic geometry that make only incidental use of category theory (e.g., look through Griffiths & Harris). | |
| Jul 16, 2023 at 16:39 | comment | added | Paul Taylor | @SBrian. I have been arguing similar things ever since MO started. But this request is not just for the benefit of undergrads but also grad students and professionals in algebraic geometry and many other disciplines. Stupid thing is that I have to wait for this Question to be Closed before the system lets me vote to Reopen it. | |
| Jul 16, 2023 at 16:18 | history | became hot network question | |||
| Jul 16, 2023 at 15:35 | answer | added | Timothy Chow | timeline score: 18 | |
| Jul 16, 2023 at 14:38 | comment | added | Donu Arapura | Category theory is extremely useful in algebraic geometry, since it clarifies old notions such as moduli spaces, and suggests new concepts such as sheaves over sites and stacks. Nevertheless, I'm not sure it's the preferred language of AG. | |
| Jul 16, 2023 at 10:01 | comment | added | Paul Taylor | Rather than Close this Question, how about giving people like @Spuire the opportunity to explain their subject to the wider community? | |
| Jul 16, 2023 at 9:42 | answer | added | user492243 | timeline score: 11 | |
| Jul 16, 2023 at 8:19 | comment | added | huurd | And, secondly, the discovery of the formalism of derived categories, which allows one to study more extensively the cohomology of algebraic varieties. I let the algebraic geometers correct me if I am wrong, and/or add more reasons for which category theory lies at the heart of algebraic geometry since Grothendieck's era. | |
| Jul 16, 2023 at 8:15 | comment | added | huurd | I am far from being an expert in algebraic geometry, but from my understanding here is roughly what happened : category theory was formalised in the mid 40's in order to capture some common features in homology and cohomology in the context of algebraic topology. Then, the use of it in algebraic geometry relies on at least two (maybe more) discoveries by Grothendieck and his student Verdier. One of them is that, surpisingly, the category of sheafs of sets on a topological space (and more generally on a site) has some good properties (I let the algebraic geometers explains which ones) | |
| Jul 16, 2023 at 7:39 | comment | added | huurd | Another question one might ask is : why does sheaf theory became an important tool in algebraic geometry from the mid 50's on ? Is this an unrelated question ? | |
| Jul 16, 2023 at 7:11 | review | Close votes | |||
| Jul 16, 2023 at 18:51 | |||||
| Jul 16, 2023 at 6:39 | comment | added | Achim Krause | I would even argue that a lot of abstract algebra becomes cleaner and more conceptual when viewed through the lens of category theory. That we don't teach it that way is of course a reasonable compromise: No one should learn about abelian categories before they learn about vector spaces, for example. | |
| Jul 16, 2023 at 6:20 | history | asked | Brian | CC BY-SA 4.0 |