Timeline for why haven't certain well-researched classes of mathematical object been framed by category theory?
Current License: CC BY-SA 2.5
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 7, 2014 at 19:30 | answer | added | Martin Brandenburg | timeline score: 3 | |
| Jan 3, 2013 at 0:13 | answer | added | paul garrett | timeline score: 3 | |
| Apr 24, 2010 at 20:22 | comment | added | Pete L. Clark | A comment on the answers of Maclean and Chapman: it is one thing to import categorical ideas and constructions into classical analysis. There is already some work on this (which for now seems to require taking a rather cockeyed look at the classical material, but a lot of great innovations feel like that initially). However, what the OP asked for was to make series, products, and indefinite integrals into categories in their own right. None of the cited references do anything like this, and my point is that one has no reason to expect that this should be possible or fruitful. | |
| Apr 24, 2010 at 6:47 | answer | added | Robin Chapman | timeline score: 7 | |
| Apr 24, 2010 at 0:31 | answer | added | Roy Maclean | timeline score: 3 | |
| Apr 23, 2010 at 21:29 | comment | added | sdcvvc | To give a balanced point, some analytical concepts can be defined categorically: jstor.org/stable/2321167, maths.gla.ac.uk/~tl/glasgowpssl/banach.pdf. But doing any nontrivial computation using categories is impossible. | |
| Apr 23, 2010 at 20:16 | answer | added | Emerton | timeline score: 17 | |
| Apr 23, 2010 at 18:04 | answer | added | Reid Barton | timeline score: 42 | |
| Apr 23, 2010 at 17:57 | comment | added | Deane Yang | I'm not a category theorist, but if I were, my response would be: "Just give us more time. We'll get there." I work mainly in areas outside the influence of category theory, so I am familiar with how ineffective abstraction has been in subjects such as differential geometry and PDE's. Despite that, I am a strong believer in the power of abstraction, and I believe that some day even PDE's will fall within its spell. It's just that the current state of category theory falls far short of what's needed. | |
| Apr 23, 2010 at 17:52 | answer | added | Pete L. Clark | timeline score: 30 | |
| Apr 23, 2010 at 16:28 | comment | added | Jonas Meyer | Tim Perutz's comment reminds me of a point made in the preface of Evans's PDE book. "PDE is not a branch of functional analysis. ... [T]he insistence on an overly abstract viewpoint, and consequent ignoring of deep calculus and measure theoretic estimates, is ultimately limiting." Even functional analytifying is often too much in analysis and differential equations, let alone categorifying. | |
| Apr 23, 2010 at 15:50 | comment | added | BCnrd | @Yemon: Maybe someone whose only tool is a claw hammer would try to use it to unscrew things, despite the absurdity of even trying. The mathematical analogue is evident. | |
| Apr 23, 2010 at 15:48 | comment | added | Tim Perutz | Some version of this question is asked here on a monthly basis. The category-theorists always respond in a reasonable way, explaining that their subject, like any other, has limitations. To be blunt: no amount of adjoint-this or colimit-that will tell you whether a sequence converges or whether a PDE has a solution. Huge swathes of current mathematics depends on proving convergence and solving PDE. | |
| Apr 23, 2010 at 15:43 | comment | added | Yemon Choi | My counter-question is "to what end" or "what purpose would it solve"? To follow up on the Mark Twain quote from Andrey Rekalo, why does no one use a claw hammer to unscrew things? | |
| Apr 23, 2010 at 15:19 | comment | added | Andrey Rekalo | "To a man with a hammer, everything looks like a nail." Mark Twain | |
| Apr 23, 2010 at 15:04 | comment | added | Martin Brandenburg | is there a name for the habit to categorify everything which you see? | |
| Apr 23, 2010 at 15:02 | comment | added | Mike Shulman | One of the first things to learn about category theory is that not everything in mathematics is a category. (-: | |
| Apr 23, 2010 at 14:40 | history | asked | graveolensa | CC BY-SA 2.5 |