Timeline for Are there any non-conjugation "extendible automorphisms" in the category of finite groups?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Aug 6 at 15:02 | comment | added | HJRW | @seldon: it’s not just a matter of taste, though. Note that the category-theoretic perspective is completely useless for actually answering the question. | |
| Aug 5 at 13:31 | comment | added | seldon | There's people with the opposite taste, so..! Glad we agree is fine then. | |
| Aug 4 at 11:37 | comment | added | HJRW | Anyway, I don't have a "case" -- I just want to advertise the fact that, contrary to initial appearances, this is an elementary and natural question! Clearly there are people who like this formulation, which is fine, but as Andy Putman's comment demonstrates, some others will be put off by it. | |
| Aug 4 at 11:10 | comment | added | HJRW | @seldon: the elementary reformulation is implicit in Dave Benson’s answer. Perhaps I will write it down when I have time, but I am very busy right now. With respect, Yuan’s reformulation is entirely content-free. Note that it doesn’t even do enough to handle the result for all groups. (“I guess we appeal to some known fact…” is where the work is!) | |
| Aug 4 at 7:54 | comment | added | seldon | @HJRW perhaps it'd help your case if you could show how the question is concisevely reformulated by excising categorical language. In any case, the fact the question can be formulated as such doesn't meant a categorical POV might not be helpful to answer of further investigate it. In fact Yuan's reformulation already makes it apparent the connection to Tannaka reconstruction (ncatlab.org/nlab/show/Tannaka+duality). | |
| Aug 3 at 14:47 | comment | added | Alec Rhea | If this was a forum aimed at undergraduate students I’d agree with the previous two comments, but… we are where we are, and the parlance used in this post isn’t anything beyond a first class in category theory. I see no issue with it. | |
| Aug 3 at 12:34 | comment | added | Andy Putman | I’m significantly more category-pilled than HJRW, but I agree with him that this question is phrased in an unnecessarily obscure way. When I first saw it, my initial impression was that it was too technical to bother trying to understand it, and I only made the effort after seeing his comment. | |
| Aug 2 at 20:58 | vote | accept | Noah Schweber | ||
| Aug 2 at 21:45 | |||||
| Aug 2 at 20:55 | comment | added | HJRW | @QiaochuYuan: in our undergraduate programme, students learn group theory in their first term and category theory in their masters or never. The question can be stated in a natural and elementary way that any undergraduate can understand. In this sense, the category-theoretic formulation is unnecessary, and also dramatically reduces the audience that can understand the question. The OP should of course do what they like; I mainly want to make the point that the statement is a great deal more elementary than the question makes it appear. | |
| Aug 2 at 17:51 | comment | added | Qiaochu Yuan | These theorems are sort of "Cayley's theorems" in the sense that they tell us that we have already found all "natural operations" of a certain kind, e.g. endomorphisms of the additive group $\mathbb{G}_a$ are "natural maps on $K$-algebras respecting addition," and endomorphisms of the forgetful functor for Banach algebras are "natural maps on Banach algebras." Try even stating these results without the category-theoretic language, it's very cumbersome! | |
| Aug 2 at 17:50 | comment | added | Qiaochu Yuan | @HJRW: why "unnecessary"? Phrasing the question in terms of endomorphisms of functors is very natural and similar questions in other parts of mathematics have well-understood and important answers. For example: endomorphisms of the additive group $\mathbb{G}_a$, regarded as its functor of points, over a field $K$ of positive characteristic $p$ correspond to polynomials in the Frobenius map. Endomorphisms of the forgetful functor from Banach algebras to sets correspond to holomorphic functions; this is a concise statement of a converse to the holomorphic functional calculus. | |
| Aug 2 at 13:22 | comment | added | HJRW | This is a beautiful question with a beautiful answer! The beauty is somewhat obscured by the unnecessary category-theoretic language, which I guess increases the length by a factor of at least 2. | |
| Aug 2 at 12:14 | history | became hot network question | |||
| Aug 2 at 7:16 | answer | added | Dave Benson | timeline score: 17 | |
| Aug 2 at 6:49 | answer | added | Qiaochu Yuan | timeline score: 11 | |
| Aug 2 at 4:14 | history | asked | Noah Schweber | CC BY-SA 4.0 |