Timeline for To derive or not to derive, that is the question
Current License: CC BY-SA 3.0
21 events
| when toggle format | what | by | license | comment | |
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| Apr 16, 2017 at 9:35 | comment | added | Ronnie Brown | To me a curious aspect of $\infty$-category theory is that it plays down the idea of composition, which is a feature of the usual category theory. So I have long been attracted by the cubical theory, which has a natural formulation of multiple compositions, a technique useful for some local-to-global problems, as I explain in several places. It is a useful research method to speculate on the possible obstructions to wider applications. If such are found to exist, that will be interesting. If they are gradually overcome ....! | |
| Apr 15, 2017 at 21:07 | comment | added | Artur Jackson | I hear from many intelligent well-known individuals to essentially "always derive." I think they don't literally mean always, but I am curious about situations in which over-using this philosophy leads to frameworks that become unsuitable for specific problems. I don't understand the amount of irritation caused by simply wondering what happens when one takes this paradigm to its logical extreme. | |
| Apr 15, 2017 at 21:01 | comment | added | Artur Jackson | I learned algebraic geometry 1-categorically, from EGA and other familiar sources. Not sure why I need to speak to anyone about the accomplishments of 1-categorical algebraic geometry; I'm well aware of Artin's work (I use it! and I do so 1-categorically). I'm curious precisely about the times at which one can argue that $\infty$-categorical or excessive use of what one would call deriving leads to an actual limiting of what one could do if done 1-categorically/non-derived/etc. | |
| Apr 15, 2017 at 11:58 | answer | added | Jason Starr | timeline score: 8 | |
| Apr 15, 2017 at 4:11 | comment | added | nfdc23 | Please speak with some experienced algebraic geometry faculty in your department; nearly all of the material in algebraic geometry books and journal articles has nothing whatsoever to do with $\infty$-categorical methods (and Artin proved deep theorems on Artin stacks perfectly well without such things). It doesn't make sense why you should feel the need to keep explaining why you aren't doing things $\infty$-categorically; do whatever is appropriate to a problem at hand, and ignore any pressure to drink the Kool Aid if you can make do with water (which is better for your health too). | |
| Apr 15, 2017 at 2:48 | comment | added | Theo Johnson-Freyd | It is of course fairly common that the passage from the strict category to the derived category is a loss of information. I find it interesting that QCoh(P^1) is very far from being a category of modules for a unital associative ring. The derived category is famously the modules of a quiver algebra. | |
| Apr 14, 2017 at 21:09 | comment | added | Artur Jackson | I also think something is being missed here. Certainly 1-categories are examples of $\infty$-categories (take the nerve), but there is a different yoga/mindset when doing fully derived versions of anything. | |
| Apr 14, 2017 at 21:04 | comment | added | Artur Jackson | @nfdc23, I think we are on the same page?? I mostly do everything 1-categorically. I often find my self explaining why I didn't do it $\infty$-categorically though. So I've been collecting some concrete examples of time when non-derived versions were actually better suited to a problem than their derived counterparts. Most of my examples are coming from $\mathbb{F}_1$-geometry. But I want to hear more. | |
| Apr 14, 2017 at 17:54 | comment | added | nfdc23 | The first paragraph is fine, but the "Motivation" (which was what I first read) is surreal; e.g., EGA is "1-categorical" and has vast depth! Not only is it good to write up explanations of things in terms that more people can understand, but if you think about how to explain ideas in as concrete a form as possible then you will come to a better understanding of where the real substance lies underneath a lot of abstract nonsense (e.g., articulate exactly why certain heavier machines are serving a real purpose in a given setting, if so). Talk with algebraic geometers in your department. | |
| Apr 14, 2017 at 16:04 | comment | added | Artur Jackson | Really I'm after: when setting up categorical geometry frameworks, how to motivate the choice for working only 1-categorically. Does it actually have merit? Apologies for cringeworthy ill-posed meta-math questions :) | |
| Apr 14, 2017 at 15:58 | comment | added | Artur Jackson | I totally agree about the badness of that phrase. Right, I like that variant quite well. I really want to ask something completely meaningless, but morally has value (I think): When are real life situations where one runs into trouble from a framework being "too derived"? (See my example above.) | |
| Apr 14, 2017 at 15:55 | comment | added | Andy Putman | The phrase "will not work" is weird; for instance, $1$-category theory is part of $\infty$-category theory. I think a better thing to ask is for topics on which the derived point of view does not seem to shed light. But this question almost answers itself: the vast majority of e.g. algebraic geometry and representation theory does not use derived techniques (at least in the sense you're describing -- of course, derived functors are everywhere, but in most cases it is fine to just work with them in a classical manner). | |
| Apr 14, 2017 at 15:54 | comment | added | Artur Jackson | Possibly another type of example is Connes and Consani's use $\Gamma$-sets. Here they use a standard stable model structure on that category (actually on $\Gamma$-spaces), but I feel this is not capturing the type of behaviour ($\mathbb{F}_1$, positivity, semi-groups) I think they are really after. I guess this could be labeled "too derived"? Already $\Gamma$-sets behaves like an "underived" version of $\Gamma$-spaces $\simeq Fun(\Delta^{op},\Gamma\mathrm{-Sets})$ | |
| Apr 14, 2017 at 15:53 | comment | added | Artur Jackson | To ask the question without sounding fluffy is a bit hard, but I think I've ran into instances where I was setting something up in a derived setting and then found myself backpedalling to get things to work in a way that I'd like. (Maybe nothing was ever "wrong" per se.) | |
| Apr 14, 2017 at 15:47 | comment | added | Artur Jackson | Thank you Ronnie, I think this is feels like the type of information I'm after. | |
| Apr 14, 2017 at 15:44 | comment | added | Ronnie Brown | In the paper arXiv:1610.07421v4 "Modelling and Computing Homotopy Types: I" I explain how the use of certain strict higher homotopy groupoids has led to a new approach to algebraic topology at the border between homology and homotopy and to new computational procedures. Work on (weak) $\infty$-groupoids has not to my knowledge yielded corresponding results. | |
| Apr 14, 2017 at 15:43 | comment | added | Artur Jackson | @CharlesRezk, I think this is the question. What are situations when it is clearly the "right move" to not worked in derived setting (whatever that may be). I'm curious of this, because I usually work 1-categorically, but sometimes find I'm explaining myself why I made this choice (rather than just doing everything $\infty$-categorically from the get go). | |
| Apr 14, 2017 at 15:40 | comment | added | Charles Rezk | What do you mean by "the derived variant will not work"? Work for what? | |
| Apr 14, 2017 at 15:26 | comment | added | Artur Jackson | @nfdc23, is this an absurd question? I've asked this to more than a few people and have heard different answers. | |
| Apr 14, 2017 at 15:17 | comment | added | nfdc23 | Where is the Kool-Aid? | |
| Apr 14, 2017 at 14:02 | history | asked | Artur Jackson | CC BY-SA 3.0 |