Timeline for candidate for rigorous _mathematical_ definition of "canonical"?
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18 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Jul 9, 2010 at 23:57 | comment | added | Konrad Waldorf | @Qiaochu: Suppose C and D are categories. There should be a difference between "there exists a functor from C to D" and "there is a canonical functor from C to D". | |
| Apr 4, 2010 at 19:37 | comment | added | Kevin Buzzard | OK, so, @Qiaochu, the answer to the question you asked above is "no" :-) But somehow it all depends on how functorial you want it, somehow. I wish I understood Francois' answer better. | |
| Apr 4, 2010 at 11:24 | comment | added | JBorger | Here's a nice little example. I'd say that the center of a group a canonical construction, but it does not prolong to a functor on the category of groups. | |
| Apr 3, 2010 at 7:58 | vote | accept | Kevin Buzzard | ||
| Apr 3, 2010 at 3:03 | comment | added | Emerton | Dear Kevin, I don't have my copy of Katz with me, but his splitting is something like splitting $H^1_{dR}$ of an elliptic curve as a sum of $H^{0,1}$ and $H^{1,0}$, a splitting which is canonical, but not functorial in the sense that it does not behave well in families. (The Hodge filtration behaves well in families, but not the Hodge decomposition, since the $H^{0,1}$ part varies anti-holomorphically, rather than holomorphically.) | |
| Apr 2, 2010 at 18:52 | comment | added | BCnrd | @Dmitri: When writing my comments I had in mind what you say, but we could always go a step further and pass to the discrete subcategory (with only identity morphisms) to make "everything" canonical. :) So I thought the main interesting feature of those little examples I mentioned was that one can make natural useful constructions whose functoriality is very restricted compared with where we begin (surjections of appropriate sort for the unipotent radical and orthogonal complement examples, etc.) | |
| Apr 2, 2010 at 18:31 | comment | added | Dmitri Pavlov | @Brian: The orthogonal complement of a closed subspace of a Hilbert space is canonical and functorial for unitary linear maps that are surjective on the subspace. Generally speaking, every instance of the word “canonical” implies that you have to choose a right category to obtain functoriality. I presume that in Katz' example one also needs to choose a different category before the splitting becomes canonical. | |
| Apr 2, 2010 at 16:50 | comment | added | BCnrd | @Kevin: Am in Michigan now, so no access to Katz' stuff; let me then answer by giving unrelated instance: unipotent radical of a smooth connected affine group over an alg. closed field is canonical (in any reasonable sense) but not functorial in the category of all such groups...though functorial with respect to surjections (not obvious just from definitions!). And orthogonal complement of a closed subspace of a Hilbert space is canonical but not functorial for (bounded) linear maps. Etc. | |
| Apr 2, 2010 at 15:32 | comment | added | Kevin Buzzard | Ps @Brian: do you understand Katz' "canonical but not functorial" statement? I bet my bottom pound that you've thought about it! | |
| Apr 2, 2010 at 15:20 | comment | added | Kevin Buzzard | Brian: of course you're right. In some sense you're raising the question as to whether one even needs to give a definition. The only thing I was flagging was that Messing was seemingly offering a reference for a mathematical definition, which somehow surprised me. You and I were both in his Paris lecture last week, right? We should have asked him! | |
| Apr 2, 2010 at 15:15 | comment | added | Will Jagy | Manjul Bhargava got a big laugh in one of his early talks on his higher composition laws. In the difficult degree five case, someone asked if the number field entity Manjul constructed had a certain desirable property, and he replied "No, but it's unique." en.wikipedia.org/wiki/Manjul_Bhargava | |
| Apr 2, 2010 at 15:15 | comment | added | BCnrd | In the spirit of the old nonsense about a tree which falls in the forest when nobody is around to hear the noise, if a mathematical concept is defined in a place which is known to almost nobody then the definition may as well not exist. Or in the words of a US Supreme Court case struggling with another widely used undefined word, as long as we know it when we see it then probably there's no need for a rigorous definition (much like there's no need for rigorous foundations of category theory as long as it is being used in an essentially linguistic or "plug example into machine" manner). | |
| Apr 2, 2010 at 12:19 | answer | added | François G. Dorais | timeline score: 12 | |
| Apr 2, 2010 at 10:34 | comment | added | Kevin Buzzard | @Qiaochu: I have in my hand a paper by Nick Katz ("p-adic properties of modular schemes and modular forms") where he says that a certain exact sequence coming from the theory of elliptic curves has a "canonical, but not functorial, splitting" (page Ka-95, a.k.a. p163 of the book). | |
| Apr 2, 2010 at 10:23 | comment | added | Qiaochu Yuan | There was one underlying mathematical definition given in the other thread, which was "functorial." Is this unsatisfying? | |
| Apr 2, 2010 at 10:22 | answer | added | Thomas Sauvaget | timeline score: 4 | |
| Apr 2, 2010 at 10:00 | history | asked | Kevin Buzzard | CC BY-SA 2.5 |